\(\int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [411]

Optimal result

Integrand size = 34, antiderivative size = 1294 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x)^3}{b^3 d}-\frac {(e+f x)^3}{b d}-\frac {a^4 (e+f x)^3}{b^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}+\frac {6 a^3 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {3 a^4 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac {3 a^4 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^4}-\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 a^2 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^4}-\frac {3 a^4 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^3 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^4 (e+f x)^3 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \]

[Out]

-a^4*(f*x+e)^3/b^3/(a^2+b^2)/d+a*(f*x+e)^3*sech(d*x+c)/b^2/d+a^2*(f*x+e)^3*tanh(d*x+c)/b^3/d-a^3*(f*x+e)^3*sec
h(d*x+c)/b^2/(a^2+b^2)/d-a^4*(f*x+e)^3*tanh(d*x+c)/b^3/(a^2+b^2)/d-a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2
)^(1/2)))/b/(a^2+b^2)^(3/2)/d+a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d-6*a^3*f
^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^3-6*I*a*f^2*(f*x+e)*polylog(2,I*ex
p(d*x+c))/b^2/d^3-6*I*a^3*f^3*polylog(3,I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+6*a^3*f*(f*x+e)^2*arctan(exp(d*x+c))/b
^2/(a^2+b^2)/d^2+3*a^4*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^3/(a^2+b^2)/d^3+3*a^4*f*(f*x+e)^2*ln(1+exp(2*d
*x+2*c))/b^3/(a^2+b^2)/d^2-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^
2+3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^2+6*a^3*f^2*(f*x+e)*polyl
og(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^3-(f*x+e)^3*tanh(d*x+c)/b/d+1/4*(f*x+e)^4/b/f-(f*x
+e)^3/b/d-6*I*a^3*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^3+6*I*a*f^2*(f*x+e)*polylog(2,-I*exp(d*
x+c))/b^2/d^3+6*I*a^3*f^3*polylog(3,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+6*I*a^3*f^2*(f*x+e)*polylog(2,I*exp(d*x+c
))/b^2/(a^2+b^2)/d^3+3/2*a^2*f^3*polylog(3,-exp(2*d*x+2*c))/b^3/d^4+a^2*(f*x+e)^3/b^3/d-3/2*f^3*polylog(3,-exp
(2*d*x+2*c))/b/d^4+6*I*a*f^3*polylog(3,I*exp(d*x+c))/b^2/d^4-6*a*f*(f*x+e)^2*arctan(exp(d*x+c))/b^2/d^2-3*a^2*
f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^3/d^3-3/2*a^4*f^3*polylog(3,-exp(2*d*x+2*c))/b^3/(a^2+b^2)/d^4-3*a^2*
f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b^3/d^2-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3
/2)/d^4+6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^4-6*I*a*f^3*polylog(3,-I*ex
p(d*x+c))/b^2/d^4+3*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b/d^2+3*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b/d^3

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 1294, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {5700, 3801, 3799, 2221, 2611, 2320, 6724, 32, 5686, 5559, 4265, 5702, 4269, 5692, 3403, 2296, 6744, 6874} \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3 a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \tanh (c+d x) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3 \text {sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 a^2}{b^3 d}-\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^2}{b^3 d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^2}{b^3 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^2}{2 b^3 d^4}+\frac {(e+f x)^3 \tanh (c+d x) a^2}{b^3 d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right ) a}{b^2 d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a}{b^2 d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a}{b^2 d^4}+\frac {(e+f x)^3 \text {sech}(c+d x) a}{b^2 d}+\frac {(e+f x)^4}{4 b f}-\frac {(e+f x)^3}{b d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^4}-\frac {(e+f x)^3 \tanh (c+d x)}{b d} \]

[In]

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

[Out]

(a^2*(e + f*x)^3)/(b^3*d) - (e + f*x)^3/(b*d) - (a^4*(e + f*x)^3)/(b^3*(a^2 + b^2)*d) + (e + f*x)^4/(4*b*f) -
(6*a*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*d^2) + (6*a^3*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)
*d^2) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f
*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (3*a^2*f*(e + f*x)^2*Log[1 + E
^(2*(c + d*x))])/(b^3*d^2) + (3*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*d^2) + (3*a^4*f*(e + f*x)^2*Log[1 +
E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^2) + ((6*I)*a*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^3) - ((
6*I)*a^3*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - ((6*I)*a*f^2*(e + f*x)*PolyLog[2,
I*E^(c + d*x)])/(b^2*d^3) + ((6*I)*a^3*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (3*a^
3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (3*a^3*f*(e
+ f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) - (3*a^2*f^2*(e + f*x
)*PolyLog[2, -E^(2*(c + d*x))])/(b^3*d^3) + (3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (3*a^4*f^
2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^3) - ((6*I)*a*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(
b^2*d^4) + ((6*I)*a^3*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) + ((6*I)*a*f^3*PolyLog[3, I*E^(c
+ d*x)])/(b^2*d^4) - ((6*I)*a^3*f^3*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) + (6*a^3*f^2*(e + f*x)*P
olyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (6*a^3*f^2*(e + f*x)*PolyLog[
3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*a^2*f^3*PolyLog[3, -E^(2*(c + d*x
))])/(2*b^3*d^4) - (3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^4) - (3*a^4*f^3*PolyLog[3, -E^(2*(c + d*x))])/(
2*b^3*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)
*d^4) + (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (a*(e + f
*x)^3*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)^3*Sech[c + d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^3*Tanh[c +
d*x])/(b^3*d) - ((e + f*x)^3*Tanh[c + d*x])/(b*d) - (a^4*(e + f*x)^3*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 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 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 3403

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

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 4269

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

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(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 5686

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*Sech
[c + d*x]*(Tanh[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]

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 5700

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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 5702

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1),
x], x] - Dist[a/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(Tanh[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 6744

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

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)^3 \tanh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int (e+f x)^3 \, dx}{b}+\frac {(3 f) \int (e+f x)^2 \tanh (c+d x) \, dx}{b d} \\ & = -\frac {(e+f x)^3}{b d}+\frac {(e+f x)^4}{4 b f}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}+\frac {a^2 \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac {(3 a f) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b^2 d}+\frac {(6 f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b d} \\ & = -\frac {(e+f x)^3}{b d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^3 \int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{b^3 d}+\frac {\left (6 i a f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 d^2}-\frac {\left (6 i a f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b d^2} \\ & = \frac {a^2 (e+f x)^3}{b^3 d}-\frac {(e+f x)^3}{b d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^3 \int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (6 a^2 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^3 d}-\frac {\left (6 i a f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{b^2 d^3}+\frac {\left (6 i a f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{b^2 d^3}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{b d^3} \\ & = \frac {a^2 (e+f x)^3}{b^3 d}-\frac {(e+f x)^3}{b d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}-\frac {3 a^2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {a^4 \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {a^3 \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (6 a^2 f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^3 d^2}-\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}+\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b d^4} \\ & = \frac {a^2 (e+f x)^3}{b^3 d}-\frac {(e+f x)^3}{b d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}-\frac {3 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^4}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^4 (e+f x)^3 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}-\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}+\frac {\left (3 a^4 f\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2 f^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{b^3 d^3} \\ & = \frac {a^2 (e+f x)^3}{b^3 d}-\frac {(e+f x)^3}{b d}-\frac {a^4 (e+f x)^3}{b^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^4}{4 b f}-\frac {6 a f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d^2}+\frac {6 a^3 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^4}+\frac {a (e+f x)^3 \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^4 (e+f x)^3 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (6 a^4 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 a^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 i a^3 f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (6 i a^3 f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (3 a^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b^3 d^4} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.13 (sec) , antiderivative size = 1111, normalized size of antiderivative = 0.86 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )}{4 b}-\frac {f \left (12 b d^3 e^2 e^{2 c} x-12 b d^3 e^2 \left (1+e^{2 c}\right ) x-12 b d^3 e f x^2-4 b d^3 f^2 x^3+12 a d^2 e^2 \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+6 b d^2 e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+12 i a d e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+6 b d e \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )+6 i a \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )\right )+b \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right )-6 d x \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}+\frac {a^3 \left (2 d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x) (a-b \text {sech}(c) \sinh (d x))}{\left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(4*b) - (f*(12*b*d^3*e^2*E^(2*c)*x - 12*b*d^3*e^2*(1 + E^(2*c)
)*x - 12*b*d^3*e*f*x^2 - 4*b*d^3*f^2*x^3 + 12*a*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 6*b*d^2*e^2*(1 + E
^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*a*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*b*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)*a*(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)]) + b*(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))])))/(2
*(a^2 + b^2)*d^4*(1 + E^(2*c))) + (a^3*(2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 3*d^3*e^2*f*x
*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + S
qrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(
c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 3*
d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/
(-a + Sqrt[a^2 + b^2])] + 6*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*e*f^2*PolyLog[3,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6
*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b
^2]))]))/(b*(a^2 + b^2)^(3/2)*d^4) + ((e + f*x)^3*Sech[c + d*x]*(a - b*Sech[c]*Sinh[d*x]))/((a^2 + b^2)*d)

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)^2/(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. 7331 vs. \(2 (1195) = 2390\).

Time = 0.40 (sec) , antiderivative size = 7331, normalized size of antiderivative = 5.67 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

Integral((e + f*x)**3*sinh(c + d*x)*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x)

Maxima [F]

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

[In]

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

[Out]

-3*b*e^2*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) - 6*a*f^3*integrate(x^
2*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 6*b*f^3*integrate(x^2/(a^2
*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 12*a*e*f^2*integrate(x*e^(d*x + c)/(a^2*d*e^
(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 12*b*e*f^2*integrate(x/(a^2*d*e^(2*d*x + 2*c) + b
^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - (a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) -
a + sqrt(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^2 + b^2)*e^(
-2*d*x - 2*c))*d) - (d*x + c)/(b*d))*e^3 - 6*a*e^2*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + 1/4*(24*b^2*e^2*f
*x + (a^2*d*f^3 + b^2*d*f^3)*x^4 + 4*(a^2*d*e*f^2 + (d*e*f^2 + 2*f^3)*b^2)*x^3 + 6*(a^2*d*e^2*f + (d*e^2*f + 4
*e*f^2)*b^2)*x^2 + ((a^2*d*f^3*e^(2*c) + b^2*d*f^3*e^(2*c))*x^4 + 4*(a^2*d*e*f^2*e^(2*c) + b^2*d*e*f^2*e^(2*c)
)*x^3 + 6*(a^2*d*e^2*f*e^(2*c) + b^2*d*e^2*f*e^(2*c))*x^2)*e^(2*d*x) + 8*(a*b*f^3*x^3*e^c + 3*a*b*e*f^2*x^2*e^
c + 3*a*b*e^2*f*x*e^c)*e^(d*x))/(a^2*b*d + b^3*d + (a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d*x)) - integrate(-2
*(a^3*f^3*x^3*e^c + 3*a^3*e*f^2*x^2*e^c + 3*a^3*e^2*f*x*e^c)*e^(d*x)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e
^(2*c))*e^(2*d*x) - 2*(a^3*b*e^c + a*b^3*e^c)*e^(d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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