\(\int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [492]

Optimal result

Integrand size = 34, antiderivative size = 1219 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {2 i b f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {b^4 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3} \]

[Out]

2*b*f^2*polylog(2,-exp(d*x+c))/a^2/d^3-2*b*f^2*polylog(2,exp(d*x+c))/a^2/d^3-1/2*b^2*f^2*polylog(3,exp(2*d*x+2
*c))/a^3/d^3+1/2*f^2*x^2/a/d-2*I*b^3*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+f*(f*x+e)*polylog(2,-
exp(2*d*x+2*c))/a/d^2-f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2+2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a/d-1/2*f^
2*polylog(3,-exp(2*d*x+2*c))/a/d^3+4*b*f*(f*x+e)*arctanh(exp(d*x+c))/a^2/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(
d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3
/(a^2+b^2)/d^2-2*I*b*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a^2/d^2-2*I*b^3*f^2*polylog(3,-I*exp(d*x+c))/a^2/(a^2+
b^2)/d^3+e*f*x/a/d-b^2*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+2*I*b*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^
2/d^2+2*I*b^3*f^2*polylog(3,I*exp(d*x+c))/a^2/(a^2+b^2)/d^3+2*I*b^3*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a^2/(a^
2+b^2)/d^2+b^4*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^2+2*I*b*f^2*polylog(3,-I*exp(d*x+c))/a^2/d
^3+2*b*(f*x+e)^2*arctan(exp(d*x+c))/a^2/d-2*b^2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^3/d+1/2*b^2*f^2*polylog(3,
-exp(2*d*x+2*c))/a^3/d^3+1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3+b^4*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b
^2)/d-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+
(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-1/2*(f*x+e)^2*coth(d*x+c)^2/a/d-f*(f*x+e)*coth(d*x+c)/a/d^2+b*(f*x+e)^2*csch
(d*x+c)/a^2/d+b^2*f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^3/d^2+f^2*ln(sinh(d*x+c))/a/d^3-2*b^3*(f*x+e)^2*arctan
(exp(d*x+c))/a^2/(a^2+b^2)/d-1/2*b^4*f^2*polylog(3,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^3+2*b^4*f^2*polylog(3,-b*e
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^3+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a
^2+b^2)/d^3-2*I*b*f^2*polylog(3,I*exp(d*x+c))/a^2/d^3

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 1219, normalized size of antiderivative = 1.00, number of steps used = 71, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {5708, 2700, 14, 5570, 6873, 12, 6874, 3801, 3556, 2631, 4267, 2611, 2320, 6724, 2701, 327, 213, 5313, 4265, 2317, 2438, 5569, 5692, 5680, 2221, 3799} \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right ) b}{a^2 d}+\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {(e+f x)^2 \text {csch}(c+d x) b}{a^2 d}+\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b}{a^2 d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b}{a^2 d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right ) b}{a^2 d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b}{a^2 d^3}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {e f x}{a d}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3} \]

[In]

Int[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) + (2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e + f*x)^2*ArcTan[
E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (2*(e + f*x)^2*ArcTanh[
E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) - (f*(e + f*x)*Coth[c + d*x])/(
a*d^2) - ((e + f*x)^2*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^2*Csch[c + d*x])/(a^2*d) - (b^4*(e + f*x)^2*Log[
1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (b^4*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (f
^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - ((2*I)*b*f*(e + f*x)*PolyLog[2
, (-I)*E^(c + d*x)])/(a^2*d^2) + ((2*I)*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) +
((2*I)*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - ((2*I)*b^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/
(a^2*(a^2 + b^2)*d^2) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d^
2) + (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) - (b^2*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^3*
d^2) - (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^3*
d^2) + ((2*I)*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - ((2*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^
2*(a^2 + b^2)*d^3) - ((2*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + ((2*I)*b^3*f^2*PolyLog[3, I*E^(c + d*
x)])/(a^2*(a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)
*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) - (b^4*f^2*Poly
Log[3, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^3) - (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) + (b^2*f^2*Po
lyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (b^2*f^2*PolyLog[3, E^
(2*c + 2*d*x)])/(2*a^3*d^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2631

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1)*(Log[u]/(b*(m + 1))), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[(a + b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5313

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan[
u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 + u^2)), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5692

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5708

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(2 f) \int (e+f x) \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a} \\ & = \frac {b (e+f x)^2 \arctan (\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(2 f) \int \frac {(e+f x) \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right )}{2 d} \, dx}{a}+\frac {(2 b f) \int (e+f x) \left (-\frac {\arctan (\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2} \\ & = \frac {b (e+f x)^2 \arctan (\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(2 b f) \int \frac {(e+f x) (-\arctan (\sinh (c+d x))-\text {csch}(c+d x))}{d} \, dx}{a^2}-\frac {f \int (e+f x) \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right ) \, dx}{a d} \\ & = \frac {b^4 (e+f x)^3}{3 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^2 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b^3 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {f \int \left (-\left ((e+f x) \coth ^2(c+d x)\right )-2 (e+f x) \log (\tanh (c+d x))\right ) \, dx}{a d}+\frac {(2 b f) \int (e+f x) (-\arctan (\sinh (c+d x))-\text {csch}(c+d x)) \, dx}{a^2 d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d} \\ & = \frac {b^4 (e+f x)^3}{3 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^2 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x)^2 \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \int (e+f x) \coth ^2(c+d x) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log (\tanh (c+d x)) \, dx}{a d}+\frac {(2 b f) \int (-((e+f x) \arctan (\sinh (c+d x)))-(e+f x) \text {csch}(c+d x)) \, dx}{a^2 d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}-\frac {\left (b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) \, dx}{a^3 d^2} \\ & = -\frac {2 b^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b (e+f x)^2 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\int 2 d (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {f \int (e+f x) \, dx}{a d}-\frac {(2 b f) \int (e+f x) \arctan (\sinh (c+d x)) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \text {csch}(c+d x) \, dx}{a^2 d}+\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {f^2 \int \coth (c+d x) \, dx}{a d^2}+\frac {\left (2 b^4 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (2 b^4 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2} \\ & = \frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}+\frac {b \int d (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 b^4 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 b f^2\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (2 b f^2\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i b^3 f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {\left (2 i b^3 f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2} \\ & = \frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2}+\frac {(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (b^4 f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2} \\ & = \frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {(2 i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {(2 i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d}-\frac {\left (b^4 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) \, dx}{a d^2}+\frac {f^2 \int \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) \, dx}{a d^2} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2784\) vs. \(2(1219)=2438\).

Time = 10.38 (sec) , antiderivative size = 2784, normalized size of antiderivative = 2.28 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

Integrate[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(b*(e + f*x)^2*Csch[c])/(a^2*d) - ((e + f*x)^2*Csch[(c + d*x)/2]^2)/(8*a*d) + (-12*a*d^3*e^2*E^(2*c)*x + 12*a*
d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)]
- 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*d*e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*
E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) - 6*a*d*e*(
1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]) + (6*I)*b*(1 + E^(2*c))
*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c + d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2
*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3, I*E^(c + d*x)]) - a*(1 + E^(2*c
))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^
(2*(c + d*x))]))/(6*(a^2 + b^2)*d^3*(1 + E^(2*c))) + (12*a^2*d^3*e^2*E^(2*c)*x - 12*b^2*d^3*e^2*E^(2*c)*x - 12
*a^2*d*E^(2*c)*f^2*x + 12*a^2*d^3*e*E^(2*c)*f*x^2 - 12*b^2*d^3*e*E^(2*c)*f*x^2 + 4*a^2*d^3*E^(2*c)*f^2*x^3 - 4
*b^2*d^3*E^(2*c)*f^2*x^3 - 24*a*b*d*e*f*ArcTanh[E^(c + d*x)] + 24*a*b*d*e*E^(2*c)*f*ArcTanh[E^(c + d*x)] + 12*
a*b*d*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d*f^2*x*Log[1 + E^(c +
d*x)] + 12*a*b*d*E^(2*c)*f^2*x*Log[1 + E^(c + d*x)] + 6*a^2*d^2*e^2*Log[1 - E^(2*(c + d*x))] - 6*b^2*d^2*e^2*
Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] + 6*b^2*d^2*e^2*E^(2*c)*Log[1 - E^(2
*(c + d*x))] - 6*a^2*f^2*Log[1 - E^(2*(c + d*x))] + 6*a^2*E^(2*c)*f^2*Log[1 - E^(2*(c + d*x))] + 12*a^2*d^2*e*
f*x*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] - 12*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^
(2*(c + d*x))] + 12*b^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] + 6*a^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))
] - 6*b^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 6*b^2*d^
2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, -E^(c + d*x)] - 12*a*b*(-1 +
E^(2*c))*f^2*PolyLog[2, E^(c + d*x)] + 6*a^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*e*f*PolyLog[2, E^(2*
(c + d*x))] - 6*a^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))
] + 6*a^2*d*f^2*x*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*f^2*x*PolyLog[2, E^(2*(c + d*x))] - 6*a^2*d*E^(2*c)*f^
2*x*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] - 3*a^2*f^2*PolyLog[3, E^(
2*(c + d*x))] + 3*b^2*f^2*PolyLog[3, E^(2*(c + d*x))] + 3*a^2*E^(2*c)*f^2*PolyLog[3, E^(2*(c + d*x))] - 3*b^2*
E^(2*c)*f^2*PolyLog[3, E^(2*(c + d*x))])/(6*a^3*d^3*(-1 + E^(2*c))) + (b^4*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^
2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b
^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(
3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) +
(6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) +
(3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*
(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*
x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*
E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*
E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*
Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^
c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^
(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])
)])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/
d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLo
g[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a
*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 +
b^2)*E^(2*c)]))])/d^3))/(3*a^3*(a^2 + b^2)*(-1 + E^(2*c))) - (a*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Csch[c]*Sech[c])
/(3*(a^2 + b^2)) + ((e + f*x)^2*Sech[(c + d*x)/2]^2)/(8*a*d) - ((e + f*x)*(-(a*f) + b*d*(e + f*x))*Csch[c/2]*C
sch[(c + d*x)/2]*Sinh[(d*x)/2])/(2*a^2*d^2) - ((e + f*x)*(a*f + b*d*(e + f*x))*Sech[c/2]*Sech[(c + d*x)/2]*Sin
h[(d*x)/2])/(2*a^2*d^2)

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3} \operatorname {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]

[In]

int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 13309 vs. \(2 (1119) = 2238\).

Time = 0.50 (sec) , antiderivative size = 13309, normalized size of antiderivative = 10.92 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**2*csch(d*x+c)**3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b
^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x -
3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) +
(a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^2 + 2*(a*f^2*x + a*e*f + (b*d*f^2*x^2*e^(3*c) + 2*b*d*e*f*x*e^(3*
c))*e^(3*d*x) - (a*d*f^2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) - (b*d*f^2*x^2*e
^c + 2*b*d*e*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) - (2*b*d*e*f +
a*f^2)*x/(a^2*d^2) + (2*b*d*e*f - a*f^2)*x/(a^2*d^2) + (2*b*d*e*f + a*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - (2
*b*d*e*f - a*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) - (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) -
2*polylog(3, -e^(d*x + c)))*(a^2*f^2 - b^2*f^2)/(a^3*d^3) - (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d
*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 - b^2*f^2)/(a^3*d^3) - 2*(a^2*d*e*f - b^2*d*e*f - a*b*f^2)*(d*x
*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) - 2*(a^2*d*e*f - b^2*d*e*f + a*b*f^2)*(d*x*log(-e^(d*x
+ c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) + 1/3*((a^2*f^2 - b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - b^2*d*e*f + a*b*
f^2)*d^2*x^2)/(a^3*d^3) + 1/3*((a^2*f^2 - b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - b^2*d*e*f - a*b*f^2)*d^2*x^2)/(a^3
*d^3) + integrate(2*(b^5*f^2*x^2 + 2*b^5*e*f*x - (a*b^4*f^2*x^2*e^c + 2*a*b^4*e*f*x*e^c)*e^(d*x))/(a^5*b + a^3
*b^3 - (a^5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^c)*e^(d*x)), x) + integrate(-2*(a*
f^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d
*x)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)