\(\int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [377]

Optimal result

Integrand size = 32, antiderivative size = 1218 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^4}{4 b f}-\frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (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 ) d}+\frac {a^2 (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 ) d}+\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {3 i a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {3 i a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 a^2 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 ) d^2}+\frac {3 a^2 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 ) d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^3}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-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 (3,i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 a^2 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 ) d^3}-\frac {6 a^2 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 ) d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^3}+\frac {3 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^3}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{b^2 d^4}-\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,-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 (4,i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^2 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 ) d^4}+\frac {6 a^2 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 ) d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b d^4}-\frac {3 a^2 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b \left (a^2+b^2\right ) d^4} \]

[Out]

a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2
)^(1/2)))/b/(a^2+b^2)/d-3/2*a^2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2+3/2*a^2*f^2*(f*x+e)*pol
ylog(3,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^3-1/4*(f*x+e)^4/b/f-3*I*a^3*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^2/(a^
2+b^2)/d^2-6*I*a^3*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b^2/(a^2+b^2)/d^3-a^2*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/b/
(a^2+b^2)/d+2*a^3*(f*x+e)^3*arctan(exp(d*x+c))/b^2/(a^2+b^2)/d-3/4*a^2*f^3*polylog(4,-exp(2*d*x+2*c))/b/(a^2+b
^2)/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+
c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^4-6*I*a*f^3*polylog(4,I*exp(d*x+c))/b^2/d^4+3/4*f^3*polylog(4,-exp(2*d*x
+2*c))/b/d^4+3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2+3*a^2*f*(f*x+e)^2*
polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/b/(a^2+b^2)/d^3-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^3-3
*I*a*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/b^2/d^2-6*I*a*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^2/d^3-6*I*a^3*f^
3*polylog(4,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+3*I*a*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^2/d^2+6*I*a*f^2*(f*x
+e)*polylog(3,I*exp(d*x+c))/b^2/d^3+6*I*a^3*f^3*polylog(4,I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+3*I*a^3*f*(f*x+e)^2*
polylog(2,I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+6*I*a^3*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^3+6*I*a
*f^3*polylog(4,-I*exp(d*x+c))/b^2/d^4+(f*x+e)^3*ln(1+exp(2*d*x+2*c))/b/d-2*a*(f*x+e)^3*arctan(exp(d*x+c))/b^2/
d+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/d^2-3/2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b/d^3

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 1218, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5700, 3799, 2221, 2611, 6744, 2320, 6724, 5686, 4265, 5692, 5680, 6874} \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^4}{4 b f}+\frac {2 a^3 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}-\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)^3}{b^2 d}+\frac {a^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}+\frac {a^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}+\frac {\log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{b d}-\frac {a^2 \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}-\frac {3 i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{b^2 d^2}+\frac {3 i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{b^2 d^2}+\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b \left (a^2+b^2\right ) d^2}+\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b \left (a^2+b^2\right ) d^2}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 b d^2}-\frac {3 a^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {6 i a^3 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{b^2 d^3}-\frac {6 i a^3 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{b^2 d^3}-\frac {6 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)}{b \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)}{b \left (a^2+b^2\right ) d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 b d^3}+\frac {3 a^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 b \left (a^2+b^2\right ) d^3}-\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,-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 (4,-i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,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 (4,i e^{c+d x}\right )}{b^2 d^4}+\frac {6 a^2 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 ) d^4}+\frac {6 a^2 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 ) d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b d^4}-\frac {3 a^2 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b \left (a^2+b^2\right ) d^4} \]

[In]

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

[Out]

-1/4*(e + f*x)^4/(b*f) - (2*a*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^2*d) + (2*a^3*(e + f*x)^3*ArcTan[E^(c + d*x)
])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) +
(a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + ((e + f*x)^3*Log[1 + E^(2
*(c + d*x))])/(b*d) - (a^2*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d) + ((3*I)*a*f*(e + f*x)^2*Po
lyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - ((3*I)*a^3*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2
)*d^2) - ((3*I)*a*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + ((3*I)*a^3*f*(e + f*x)^2*PolyLog[2, I*E
^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]
)/(b*(a^2 + b^2)*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b
^2)*d^2) + (3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b*d^2) - (3*a^2*f*(e + f*x)^2*PolyLog[2, -E^(2*(c
+ d*x))])/(2*b*(a^2 + b^2)*d^2) - ((6*I)*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^3) + ((6*I)*a^3
*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) + ((6*I)*a*f^2*(e + f*x)*PolyLog[3, I*E^(c
+ d*x)])/(b^2*d^3) - ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (6*a^2*f^2*(e
+ f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (6*a^2*f^2*(e + f*x)*PolyL
og[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(2*(c +
d*x))])/(2*b*d^3) + (3*a^2*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^3) + ((6*I)*a*f^3*Po
lyLog[4, (-I)*E^(c + d*x)])/(b^2*d^4) - ((6*I)*a^3*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) - (
(6*I)*a*f^3*PolyLog[4, I*E^(c + d*x)])/(b^2*d^4) + ((6*I)*a^3*f^3*PolyLog[4, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*
d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (6*a^2*f^3*PolyL
og[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4
*b*d^4) - (3*a^2*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b*(a^2 + b^2)*d^4)

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3251\) vs. \(2(1218)=2436\).

Time = 11.15 (sec) , antiderivative size = 3251, normalized size of antiderivative = 2.67 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-1/4*(8*b*d^4*e^3*E^(2*c)*x + 12*b*d^4*e^2*E^(2*c)*f*x^2 + 8*b*d^4*e*E^(2*c)*f^2*x^3 + 2*b*d^4*E^(2*c)*f^3*x^4
+ 8*a*d^3*e^3*ArcTan[E^(c + d*x)] + 8*a*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] + (12*I)*a*d^3*e^2*f*x*Log[1 - I*
E^(c + d*x)] + (12*I)*a*d^3*e^2*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (12*I)*a*d^3*e*f^2*x^2*Log[1 - I*E^(c + d
*x)] + (12*I)*a*d^3*e*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (4*I)*a*d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] + (4
*I)*a*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d*x)] - (12*I)*a*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*
e^2*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e*E^(2*c
)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*E^(2*c)*f^3*x^3*Lo
g[1 + I*E^(c + d*x)] - 4*b*d^3*e^3*Log[1 + E^(2*(c + d*x))] - 4*b*d^3*e^3*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 1
2*b*d^3*e^2*f*x*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e*f^2*
x^2*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - 4*b*d^3*f^3*x^3*Log[1 + E
^(2*(c + d*x))] - 4*b*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(2*(c + d*x))] - (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*
PolyLog[2, (-I)*E^(c + d*x)] + (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] - 6*b*d^2*e^
2*f*PolyLog[2, -E^(2*(c + d*x))] - 6*b*d^2*e^2*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] - 12*b*d^2*e*f^2*x*PolyL
og[2, -E^(2*(c + d*x))] - 12*b*d^2*e*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 6*b*d^2*f^3*x^2*PolyLog[2, -
E^(2*(c + d*x))] - 6*b*d^2*E^(2*c)*f^3*x^2*PolyLog[2, -E^(2*(c + d*x))] + (24*I)*a*d*e*f^2*PolyLog[3, (-I)*E^(
c + d*x)] + (24*I)*a*d*e*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (24*I)*a*d*f^3*x*PolyLog[3, (-I)*E^(c + d*
x)] + (24*I)*a*d*E^(2*c)*f^3*x*PolyLog[3, (-I)*E^(c + d*x)] - (24*I)*a*d*e*f^2*PolyLog[3, I*E^(c + d*x)] - (24
*I)*a*d*e*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (24*I)*a*d*f^3*x*PolyLog[3, I*E^(c + d*x)] - (24*I)*a*d*E^(2
*c)*f^3*x*PolyLog[3, I*E^(c + d*x)] + 6*b*d*e*f^2*PolyLog[3, -E^(2*(c + d*x))] + 6*b*d*e*E^(2*c)*f^2*PolyLog[3
, -E^(2*(c + d*x))] + 6*b*d*f^3*x*PolyLog[3, -E^(2*(c + d*x))] + 6*b*d*E^(2*c)*f^3*x*PolyLog[3, -E^(2*(c + d*x
))] - (24*I)*a*f^3*PolyLog[4, (-I)*E^(c + d*x)] - (24*I)*a*E^(2*c)*f^3*PolyLog[4, (-I)*E^(c + d*x)] + (24*I)*a
*f^3*PolyLog[4, I*E^(c + d*x)] + (24*I)*a*E^(2*c)*f^3*PolyLog[4, I*E^(c + d*x)] - 3*b*f^3*PolyLog[4, -E^(2*(c
+ d*x))] - 3*b*E^(2*c)*f^3*PolyLog[4, -E^(2*(c + d*x))])/((a^2 + b^2)*d^4*(1 + E^(2*c))) - (a^2*(4*e^3*E^(2*c)
*x + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f^3*x^4 + (4*a*Sqrt[a^2 + b^2]*e^3*ArcTan[(a + b*E^(c
+ d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (4*a*Sqrt[-a^2 - b^2]*e^3*ArcTanh[(a + b*E^(c + d*x))/S
qrt[a^2 + b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) - (2*e^3*E^(2*c)*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])/d + (
2*e^3*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[
(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*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 + (6*e*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*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 + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c -
Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*
c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*L
og[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*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*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 + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3
*x^3*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)^2*PolyL
og[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)^2*PolyLog
[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (12*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/
(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*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 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d
^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*e*f^2*P
olyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*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 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + S
qrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*
E^(2*c)]))])/d^3 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(
2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 + (12*f^3*PolyLog[4, -((b*E
^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E
^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4))/(2*b*(a^2 + b^2)*(-1 + E^(2*c))) + ((4*a^2*e^3*x - 4*b^2*e^3*x + 6*a^
2*e^2*f*x^2 - 6*b^2*e^2*f*x^2 + 4*a^2*e*f^2*x^3 - 4*b^2*e*f^2*x^3 + a^2*f^3*x^4 - b^2*f^3*x^4 + 4*a^2*e^3*x*Co
sh[2*c] + 4*b^2*e^3*x*Cosh[2*c] + 6*a^2*e^2*f*x^2*Cosh[2*c] + 6*b^2*e^2*f*x^2*Cosh[2*c] + 4*a^2*e*f^2*x^3*Cosh
[2*c] + 4*b^2*e*f^2*x^3*Cosh[2*c] + a^2*f^3*x^4*Cosh[2*c] + b^2*f^3*x^4*Cosh[2*c])*Csch[c]*Sech[c])/(8*b*(a^2
+ b^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 1962, normalized size of antiderivative = 1.61 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*e*f^2*x^3 + 6*(a^2 + b^2)*d^4*e^2*f*x^2 + 4*(a^2 + b^2)*d^4*
e^3*x - 24*a^2*f^3*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a
^2 + b^2)/b^2))/b) - 24*a^2*f^3*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x
+ c))*sqrt((a^2 + b^2)/b^2))/b) - 12*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*dilog((a*cosh(d*x +
c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 12*(a^2*d^2*f^
3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sin
h(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*(I*a*b*d^2*f^3*x^2 - b^2*d^2*f^3*x^2 + 2*I*a*b*d^2*e*f^2*x
- 2*b^2*d^2*e*f^2*x + I*a*b*d^2*e^2*f - b^2*d^2*e^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 12*(-I*a*b*d
^2*f^3*x^2 - b^2*d^2*f^3*x^2 - 2*I*a*b*d^2*e*f^2*x - 2*b^2*d^2*e*f^2*x - I*a*b*d^2*e^2*f - b^2*d^2*e^2*f)*dilo
g(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 4*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*
log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^
2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) +
2*a) - 4*(a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 +
a^2*c^3*f^3)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b
^2) - b)/b) - 4*(a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e
*f^2 + a^2*c^3*f^3)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 +
b^2)/b^2) - b)/b) + 4*(I*a*b*d^3*e^3 - b^2*d^3*e^3 - 3*I*a*b*c*d^2*e^2*f + 3*b^2*c*d^2*e^2*f + 3*I*a*b*c^2*d*e
*f^2 - 3*b^2*c^2*d*e*f^2 - I*a*b*c^3*f^3 + b^2*c^3*f^3)*log(cosh(d*x + c) + sinh(d*x + c) + I) + 4*(-I*a*b*d^3
*e^3 - b^2*d^3*e^3 + 3*I*a*b*c*d^2*e^2*f + 3*b^2*c*d^2*e^2*f - 3*I*a*b*c^2*d*e*f^2 - 3*b^2*c^2*d*e*f^2 + I*a*b
*c^3*f^3 + b^2*c^3*f^3)*log(cosh(d*x + c) + sinh(d*x + c) - I) + 4*(-I*a*b*d^3*f^3*x^3 - b^2*d^3*f^3*x^3 - 3*I
*a*b*d^3*e*f^2*x^2 - 3*b^2*d^3*e*f^2*x^2 - 3*I*a*b*d^3*e^2*f*x - 3*b^2*d^3*e^2*f*x - 3*I*a*b*c*d^2*e^2*f - 3*b
^2*c*d^2*e^2*f + 3*I*a*b*c^2*d*e*f^2 + 3*b^2*c^2*d*e*f^2 - I*a*b*c^3*f^3 - b^2*c^3*f^3)*log(I*cosh(d*x + c) +
I*sinh(d*x + c) + 1) + 4*(I*a*b*d^3*f^3*x^3 - b^2*d^3*f^3*x^3 + 3*I*a*b*d^3*e*f^2*x^2 - 3*b^2*d^3*e*f^2*x^2 +
3*I*a*b*d^3*e^2*f*x - 3*b^2*d^3*e^2*f*x + 3*I*a*b*c*d^2*e^2*f - 3*b^2*c*d^2*e^2*f - 3*I*a*b*c^2*d*e*f^2 + 3*b^
2*c^2*d*e*f^2 + I*a*b*c^3*f^3 - b^2*c^3*f^3)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + 24*(I*a*b*f^3 - b^2
*f^3)*polylog(4, I*cosh(d*x + c) + I*sinh(d*x + c)) + 24*(-I*a*b*f^3 - b^2*f^3)*polylog(4, -I*cosh(d*x + c) -
I*sinh(d*x + c)) + 24*(a^2*d*f^3*x + a^2*d*e*f^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24*(a^2*d*f^3*x + a^2*d*e*f^2)*polylog(3, (a*cosh(d*x + c)
+ a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24*(-I*a*b*d*f^3*x + b^2*
d*f^3*x - I*a*b*d*e*f^2 + b^2*d*e*f^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) + 24*(I*a*b*d*f^3*x + b^2
*d*f^3*x + I*a*b*d*e*f^2 + b^2*d*e*f^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x + c)))/((a^2*b + b^3)*d^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

e^3*(a^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b + b^3)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 +
b^2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + (d*x + c)/(b*d)) + 1/4*(f^3*x^4 + 4*e*f^2*x^3 + 6*e^2*
f*x^2)/b - integrate(2*(a^2*b*f^3*x^3 + 3*a^2*b*e*f^2*x^2 + 3*a^2*b*e^2*f*x - (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) - integrate(2*(b*f^3*x^3 + 3*b*e*f^2*x^2 + 3*b*e^2*f*x + (a*f^3*x^3*e^c + 3*a*e*f^2
*x^2*e^c + 3*a*e^2*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. \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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