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

Optimal. Leaf size=978 \[ \frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}-\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^5}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {i f \text {Li}_2\left (i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {(e+f x) \text {sech}^2(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {i f \text {Li}_2\left (i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {f \text {sech}(c+d x) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) b^2}{2 a \left (a^2+b^2\right ) d}+\frac {(e+f x) \tanh ^2(c+d x) b}{2 a^2 d}-\frac {f x b}{2 a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b}{a^2 d}+\frac {f x \log (\tanh (c+d x)) b}{a^2 d}-\frac {(e+f x) \log (\tanh (c+d x)) b}{a^2 d}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right ) b}{2 a^2 d^2}+\frac {f \tanh (c+d x) b}{2 a^2 d^2}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2} \]

[Out]

3/2*I*f*polylog(2,-I*exp(d*x+c))/a/d^2-1/2*b^5*f*polylog(2,-exp(2*d*x+2*c))/a^2/(a^2+b^2)^2/d^2+1/2*b^2*f*sech
(d*x+c)/a/(a^2+b^2)/d^2+1/2*b^3*(f*x+e)*sech(d*x+c)^2/a^2/(a^2+b^2)/d-1/2*b^3*f*tanh(d*x+c)/a^2/(a^2+b^2)/d^2+
2*b^4*(f*x+e)*arctan(exp(d*x+c))/a/(a^2+b^2)^2/d+2*b*f*x*arctanh(exp(2*d*x+2*c))/a^2/d+b^2*(f*x+e)*arctan(exp(
d*x+c))/a/(a^2+b^2)/d+I*b^4*f*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)^2/d^2+1/2*b^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c
)/a/(a^2+b^2)/d-I*b^4*f*polylog(2,-I*exp(d*x+c))/a/(a^2+b^2)^2/d^2-1/2*I*b^2*f*polylog(2,-I*exp(d*x+c))/a/(a^2
+b^2)/d^2-1/2*b*f*x/a^2/d+1/2*(f*x+e)*csch(d*x+c)*sech(d*x+c)^2/a/d+1/2*b*f*tanh(d*x+c)/a^2/d^2+1/2*b*(f*x+e)*
tanh(d*x+c)^2/a^2/d-3/2*I*f*polylog(2,I*exp(d*x+c))/a/d^2-3*f*x*arctan(exp(d*x+c))/a/d+3/2*f*x*arctan(sinh(d*x
+c))/a/d+1/2*b*f*polylog(2,-exp(2*d*x+2*c))/a^2/d^2-1/2*b*f*polylog(2,exp(2*d*x+2*c))/a^2/d^2+1/2*I*b^2*f*poly
log(2,I*exp(d*x+c))/a/(a^2+b^2)/d^2-3/2*(f*x+e)*arctan(sinh(d*x+c))/a/d-f*arctanh(cosh(d*x+c))/a/d^2-3/2*(f*x+
e)*csch(d*x+c)/a/d-1/2*f*sech(d*x+c)/a/d^2+b^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^2/
d+b*f*x*ln(tanh(d*x+c))/a^2/d+b^5*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^2/d^2+b^5*f*pol
ylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^2/d^2-b^5*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)^2
/d+b^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^2/d-b*(f*x+e)*ln(tanh(d*x+c))/a^2/d

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Rubi [A]  time = 1.41, antiderivative size = 978, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 27, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.794, Rules used = {5589, 2621, 288, 321, 207, 5462, 5203, 12, 4180, 2279, 2391, 3770, 2622, 2620, 14, 2548, 4182, 3473, 8, 5573, 5561, 2190, 6742, 3718, 4185, 5451, 3767} \[ \frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}-\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^5}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {(e+f x) \text {sech}^2(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {f \text {sech}(c+d x) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) b^2}{2 a \left (a^2+b^2\right ) d}+\frac {(e+f x) \tanh ^2(c+d x) b}{2 a^2 d}-\frac {f x b}{2 a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b}{a^2 d}+\frac {f x \log (\tanh (c+d x)) b}{a^2 d}-\frac {(e+f x) \log (\tanh (c+d x)) b}{a^2 d}+\frac {f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {f \text {PolyLog}\left (2,e^{2 c+2 d x}\right ) b}{2 a^2 d^2}+\frac {f \tanh (c+d x) b}{2 a^2 d^2}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {3 i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*f*x)/(2*a^2*d) - (3*f*x*ArcTan[E^(c + d*x)])/(a*d) + (2*b^4*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)^
2*d) + (b^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) + (3*f*x*ArcTan[Sinh[c + d*x]])/(2*a*d) - (3*(e +
f*x)*ArcTan[Sinh[c + d*x]])/(2*a*d) + (2*b*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - (f*ArcTanh[Cosh[c + d*x]])
/(a*d^2) - (3*(e + f*x)*Csch[c + d*x])/(2*a*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
)/(a^2*(a^2 + b^2)^2*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^2*d)
- (b^5*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)^2*d) + (b*f*x*Log[Tanh[c + d*x]])/(a^2*d) - (b*(e
+ f*x)*Log[Tanh[c + d*x]])/(a^2*d) + (((3*I)/2)*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (I*b^4*f*PolyLog[2,
(-I)*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) - ((I/2)*b^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (
((3*I)/2)*f*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (I*b^4*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) + (
(I/2)*b^2*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(a^2*(a^2 + b^2)^2*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 +
b^2)^2*d^2) - (b^5*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)^2*d^2) + (b*f*PolyLog[2, -E^(2*c + 2*d*
x)])/(2*a^2*d^2) - (b*f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^2*d^2) - (f*Sech[c + d*x])/(2*a*d^2) + (b^2*f*Sech[c
+ d*x])/(2*a*(a^2 + b^2)*d^2) + (b^3*(e + f*x)*Sech[c + d*x]^2)/(2*a^2*(a^2 + b^2)*d) + ((e + f*x)*Csch[c + d
*x]*Sech[c + d*x]^2)/(2*a*d) + (b*f*Tanh[c + d*x])/(2*a^2*d^2) - (b^3*f*Tanh[c + d*x])/(2*a^2*(a^2 + b^2)*d^2)
+ (b^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*a*(a^2 + b^2)*d) + (b*(e + f*x)*Tanh[c + d*x]^2)/(2*a^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 207

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

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 2620

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 2621

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 2622

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3718

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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3770

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

Rule 4180

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 4182

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 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 5203

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 + u^2), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5462

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 5561

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 5573

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 5589

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 6742

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

Rubi steps

\begin {align*} \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}-\frac {b \int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {f \int \left (-\frac {3 \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {3 \text {csch}(c+d x)}{2 d}+\frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {b^2 \int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(b f) \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a^2}-\frac {f \int \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{2 a d}+\frac {(3 f) \int \tan ^{-1}(\sinh (c+d x)) \, dx}{2 a d}+\frac {(3 f) \int \text {csch}(c+d x) \, dx}{2 a d}\\ &=\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 f \tanh ^{-1}(\cosh (c+d x))}{2 a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {b^4 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^6 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^2 \int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}-\frac {f \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a d^2}-\frac {(3 f) \int d x \text {sech}(c+d x) \, dx}{2 a d}-\frac {(b f) \int \tanh ^2(c+d x) \, dx}{2 a^2 d}+\frac {(b f) \int \log (\tanh (c+d x)) \, dx}{a^2 d}\\ &=-\frac {b^5 (e+f x)^2}{2 a^2 \left (a^2+b^2\right )^2 f}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 f \tanh ^{-1}(\cosh (c+d x))}{2 a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {b^4 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^6 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^6 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^2 \int (e+f x) \text {sech}^3(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(3 f) \int x \text {sech}(c+d x) \, dx}{2 a}-\frac {f \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a d^2}-\frac {(b f) \int 1 \, dx}{2 a^2 d}-\frac {(b f) \int 2 d x \text {csch}(2 c+2 d x) \, dx}{a^2 d}\\ &=-\frac {b f x}{2 a^2 d}-\frac {b^5 (e+f x)^2}{2 a^2 \left (a^2+b^2\right )^2 f}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {b^4 \int (e+f x) \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int (e+f x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^2 \int (e+f x) \text {sech}(c+d x) \, dx}{2 a \left (a^2+b^2\right )}-\frac {(2 b f) \int x \text {csch}(2 c+2 d x) \, dx}{a^2}+\frac {(3 i f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a d}-\frac {(3 i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a d}-\frac {\left (b^5 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (b^5 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (b^3 f\right ) \int \text {sech}^2(c+d x) \, dx}{2 a^2 \left (a^2+b^2\right ) d}\\ &=-\frac {b f x}{2 a^2 d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}-\frac {\left (2 b^5\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {(3 i f) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac {(3 i f) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac {\left (b^5 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {\left (b^5 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {\left (i b^3 f\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {(b f) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac {\left (i b^4 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (i b^4 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}-\frac {\left (i b^2 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a \left (a^2+b^2\right ) d}+\frac {\left (i b^2 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a \left (a^2+b^2\right ) d}\\ &=-\frac {b f x}{2 a^2 d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}-\frac {b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {(b f) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {(b f) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {\left (i b^4 f\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {\left (i b^4 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {\left (i b^2 f\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {\left (i b^2 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {\left (b^5 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {b f x}{2 a^2 d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}-\frac {b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac {\left (b^5 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}\\ &=-\frac {b f x}{2 a^2 d}-\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b^5 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {b^5 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}-\frac {b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}\\ \end {align*}

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Mathematica [A]  time = 10.92, size = 1337, normalized size = 1.37 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*(((-(d*e*Cosh[(c + d*x)/2]) + c*f*Cosh[(c + d*x)/2] - f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2]*Csch[
c + d*x]*(a + b*Sinh[c + d*x]))/(16*a*d^2*(b + a*Csch[c + d*x])) - (b*e*Csch[c + d*x]*Log[Sinh[c + d*x]]*(a +
b*Sinh[c + d*x]))/(8*a^2*d*(b + a*Csch[c + d*x])) + (b*c*f*Csch[c + d*x]*Log[Sinh[c + d*x]]*(a + b*Sinh[c + d*
x]))/(8*a^2*d^2*(b + a*Csch[c + d*x])) + (f*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]]*(a + b*Sinh[c + d*x]))/(8*a*d
^2*(b + a*Csch[c + d*x])) + ((I/8)*b*f*Csch[c + d*x]*(I*(c + d*x)*Log[1 - E^(-2*(c + d*x))] - (I/2)*(-(c + d*x
)^2 + PolyLog[2, E^(-2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/(a^2*d^2*(b + a*Csch[c + d*x])) + (b^5*Csch[c + d*
x]*(-1/2*(f*(c + d*x)^2) + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e*Log[a + b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + f*PolyLog
[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])*(a + b*S
inh[c + d*x]))/(8*a^2*(a^2 + b^2)^2*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*(-2*a^2*b*d*e*(c + d*x) - 4*b^
3*d*e*(c + d*x) + 2*a^2*b*c*f*(c + d*x) + 4*b^3*c*f*(c + d*x) - a^2*b*f*(c + d*x)^2 - 2*b^3*f*(c + d*x)^2 - 6*
a^3*d*e*ArcTan[E^(c + d*x)] - 10*a*b^2*d*e*ArcTan[E^(c + d*x)] + 6*a^3*c*f*ArcTan[E^(c + d*x)] + 10*a*b^2*c*f*
ArcTan[E^(c + d*x)] - (3*I)*a^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (5*I)*a*b^2*f*(c + d*x)*Log[1 - I*E^(c +
d*x)] + (3*I)*a^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + (5*I)*a*b^2*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^2*
b*d*e*Log[1 + E^(2*(c + d*x))] + 4*b^3*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^2*b*c*f*Log[1 + E^(2*(c + d*x))] - 4
*b^3*c*f*Log[1 + E^(2*(c + d*x))] + 2*a^2*b*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + 4*b^3*f*(c + d*x)*Log[1 + E
^(2*(c + d*x))] + I*a*(3*a^2 + 5*b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] - I*a*(3*a^2 + 5*b^2)*f*PolyLog[2, I*E^(c
+ d*x)] + a^2*b*f*PolyLog[2, -E^(2*(c + d*x))] + 2*b^3*f*PolyLog[2, -E^(2*(c + d*x))])*(a + b*Sinh[c + d*x]))
/(16*(a^2 + b^2)^2*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*Sech[(c + d*x)/2]*(d*e*Sinh[(c + d*x)/2] - c*f*
Sinh[(c + d*x)/2] + f*(c + d*x)*Sinh[(c + d*x)/2])*(a + b*Sinh[c + d*x]))/(16*a*d^2*(b + a*Csch[c + d*x])) + (
Csch[c + d*x]*Sech[c + d*x]*(a + b*Sinh[c + d*x])*(-(a*f) + b*f*Sinh[c + d*x]))/(16*(a^2 + b^2)*d^2*(b + a*Csc
h[c + d*x])) + (Csch[c + d*x]*Sech[c + d*x]^2*(a + b*Sinh[c + d*x])*(-(b*d*e) + b*c*f - b*f*(c + d*x) - a*d*e*
Sinh[c + d*x] + a*c*f*Sinh[c + d*x] - a*f*(c + d*x)*Sinh[c + d*x]))/(16*(a^2 + b^2)*d^2*(b + a*Csch[c + d*x]))
)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.41, size = 3280, normalized size = 3.35 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a^2/(a^2+b^2)^2*b^5*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d^2/a^2/(a^2+b^2)^2*b^5*f*dilog((-b*exp(d*x+
c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2/a^2/(a^2+b^2)^2*b^5*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)
/(a+(a^2+b^2)^(1/2)))-12/d*a^3/(a^2+b^2)*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))-1/(a^2+b^2)^(5/2)/d^2*f*b*arctanh(
1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/(a^2+b^2)^(5/2)/d^2*f*b^3*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^
2+b^2)^(1/2))*a-1/(a^2+b^2)/d^2/a^2*b^3*f*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d^2/a^2*b^3*f*dilog(exp(d*x+c))+1/(a
^2+b^2)/d^2/a*b^2*f*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d^2/a*b^2*f*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d/a^2*b^3*e*ln(exp(d
*x+c)+1)-1/(a^2+b^2)/d/a^2*b^3*e*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d^2*b*f*c*ln(exp(d*x+c)-1)+6*I/d^2*a^3/(a^2+b^2)
*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-6*I/d^2*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+20/d^2*a/(a^2
+b^2)*b^2*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))+7/2/d^2*a/(a^2+b^2)^(5/2)*b^3*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2
*a)/(a^2+b^2)^(1/2))+4/d*a^2/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x+4/d*a^2/(a^2+b^2)*f/(4*a^2+4*b^2
)*ln(1-I*exp(d*x+c))*b*x-4/d^2*a^2/(a^2+b^2)*f*c/(4*a^2+4*b^2)*b*ln(1+exp(2*d*x+2*c))+4/d^2*a^2/(a^2+b^2)*f/(4
*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c+4/d^2*a^2/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c-10*I/d^2*a/(a^2+
b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+
6*I/d*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-6*I/d*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c)
)*x+1/d^2/(a^2+b^2)^(3/2)*f*b^3/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3/2*b/d*e/(a^2+b^2)^(3/2)*
arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a+2/d^2/(a^2+b^2)^(5/2)*b^5*f*c/a*arctanh(1/2*(2*b*exp(d*x+c
)+2*a)/(a^2+b^2)^(1/2))+3/2/d^2/(a^2+b^2)^(5/2)*a^3*f*b*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/
d^2/(a^2+b^2)^(3/2)*b^3*f*c/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)/d^2/a^2*b^3*f*c*ln
(exp(d*x+c)-1)-8/d^2/(a^2+b^2)*b^3*f*c/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+1/d/a^2/(a^2+b^2)^2*b^5*f*ln((-b*exp
(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d/a^2/(a^2+b^2)^2*b^5*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+
a)/(a+(a^2+b^2)^(1/2)))*x+4/d*a^2/(a^2+b^2)*e/(4*a^2+4*b^2)*b*ln(1+exp(2*d*x+2*c))+8/d/(a^2+b^2)*b^3*f/(4*a^2+
4*b^2)*ln(1-I*exp(d*x+c))*x+8/d/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+8/d^2/(a^2+b^2)*b^3*f/(4*a^
2+4*b^2)*ln(1-I*exp(d*x+c))*c+8/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-1/(a^2+b^2)/d*ln(exp(d*
x+c)+1)*b*f*x+8/d/(a^2+b^2)*b^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+8/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog
(1+I*exp(d*x+c))+8/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4
*b^2)*ln(1+I*exp(d*x+c))*c-10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/d*a/(a^2+b^2)*b^
2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-10*I/d*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-3/2*b/d^2*f
*c/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a+1/d^2/(a^2+b^2)^(3/2)*a*f*b*arctanh(1/2
*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2/(a^2+b^2)^(5/2)*f*b^5/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2
)^(1/2))-3/2/d/(a^2+b^2)^(5/2)*a^3*b*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d/(a^2+b^2)^(5/2)*b
^5*e/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d/(a^2+b^2)^(3/2)*b^3*e/a*arctanh(1/2*(2*b*exp(d*x+
c)+2*a)/(a^2+b^2)^(1/2))+4/d^2*a^2/(a^2+b^2)*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b+4/d^2*a^2/(a^2+b^2)*f/(4*
a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b+12/d^2*a^3/(a^2+b^2)*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/d^2/a^2/(a^2+b^
2)^2*b^5*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d^2/a^2/(a^2+b^2)^2*b^5*f*ln((b*exp(
d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d^2/a^2/(a^2+b^2)^2*b^5*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x
+c)-b)-7/2/d*a/(a^2+b^2)^(5/2)*b^3*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-20/d*a/(a^2+b^2)*b^2*e/
(4*a^2+4*b^2)*arctan(exp(d*x+c))+6*I/d^2*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-6*I/d^2*a^3/(a^2+
b^2)*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-(3*a^2*d*f*x*exp(5*d*x+5*c)+2*b^2*d*f*x*exp(5*d*x+5*c)+3*a^2*d*e*ex
p(5*d*x+5*c)+2*a*b*d*f*x*exp(4*d*x+4*c)+2*b^2*d*e*exp(5*d*x+5*c)+2*a^2*d*f*x*exp(3*d*x+3*c)+a^2*f*exp(5*d*x+5*
c)+2*a*b*d*e*exp(4*d*x+4*c)+4*b^2*d*f*x*exp(3*d*x+3*c)+2*a^2*d*e*exp(3*d*x+3*c)-2*a*b*d*f*x*exp(2*d*x+2*c)+a*b
*f*exp(4*d*x+4*c)+4*b^2*d*e*exp(3*d*x+3*c)+3*a^2*d*f*x*exp(d*x+c)-2*a*b*d*e*exp(2*d*x+2*c)+2*b^2*d*f*x*exp(d*x
+c)+3*a^2*d*e*exp(d*x+c)+2*b^2*d*e*exp(d*x+c)-a^2*f*exp(d*x+c)-a*b*f)/d^2/a/(exp(2*d*x+2*c)-1)/(a^2+b^2)/(1+ex
p(2*d*x+2*c))^2-1/(a^2+b^2)/d/a^2*b^3*f*ln(exp(d*x+c)+1)*x-1/(a^2+b^2)/d^2*b*f*dilog(exp(d*x+c)+1)+1/(a^2+b^2)
/d^2*b*f*dilog(exp(d*x+c))-1/(a^2+b^2)/d*b*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d*b*e*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d
^2*a*f*ln(exp(d*x+c)+1)+1/(a^2+b^2)/d^2*a*f*ln(exp(d*x+c)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^6 + 2*a^4*b^2 + a^2*b^4)*d) + (3*a^3 + 5*a*b^2)*arcta
n(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a^2*b + 2*b^3)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^
4)*d) - (2*a*b*e^(-2*d*x - 2*c) - 2*a*b*e^(-4*d*x - 4*c) + (3*a^2 + 2*b^2)*e^(-d*x - c) + 2*(a^2 + 2*b^2)*e^(-
3*d*x - 3*c) + (3*a^2 + 2*b^2)*e^(-5*d*x - 5*c))/((a^3 + a*b^2 + (a^3 + a*b^2)*e^(-2*d*x - 2*c) - (a^3 + a*b^2
)*e^(-4*d*x - 4*c) - (a^3 + a*b^2)*e^(-6*d*x - 6*c))*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c)
- 1)/(a^2*d))*e + (32*b*d*integrate(1/32*x/(a^2*d*e^(d*x + c) + a^2*d), x) - 32*b*d*integrate(1/32*x/(a^2*d*e
^(d*x + c) - a^2*d), x) + a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) + 1)/(a^2*d^2)) - a*((d*x + c)/(a^2*d^2) -
log(e^(d*x + c) - 1)/(a^2*d^2)) - (2*a*b*d*x*e^(2*d*x + 2*c) - 2*(a^2*d*e^(3*c) + 2*b^2*d*e^(3*c))*x*e^(3*d*x)
+ a*b - (a^2*e^(5*c) + (3*a^2*d*e^(5*c) + 2*b^2*d*e^(5*c))*x)*e^(5*d*x) - (2*a*b*d*x*e^(4*c) + a*b*e^(4*c))*e
^(4*d*x) + (a^2*e^c - (3*a^2*d*e^c + 2*b^2*d*e^c)*x)*e^(d*x))/(a^3*d^2 + a*b^2*d^2 - (a^3*d^2*e^(6*c) + a*b^2*
d^2*e^(6*c))*e^(6*d*x) - (a^3*d^2*e^(4*c) + a*b^2*d^2*e^(4*c))*e^(4*d*x) + (a^3*d^2*e^(2*c) + a*b^2*d^2*e^(2*c
))*e^(2*d*x)) - 32*integrate(-1/16*(a*b^5*x*e^(d*x + c) - b^6*x)/(a^6*b + 2*a^4*b^3 + a^2*b^5 - (a^6*b*e^(2*c)
+ 2*a^4*b^3*e^(2*c) + a^2*b^5*e^(2*c))*e^(2*d*x) - 2*(a^7*e^c + 2*a^5*b^2*e^c + a^3*b^4*e^c)*e^(d*x)), x) - 3
2*integrate(1/32*((3*a^3*e^c + 5*a*b^2*e^c)*x*e^(d*x) + 2*(a^2*b + 2*b^3)*x)/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(
2*c) + 2*a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x))*f

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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