3.5.8 \(\int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [408]

Optimal. Leaf size=631 \[ \frac {a (e+f x)^2}{2 b^2 f}+\frac {2 a^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {f \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {i a^2 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2}+\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i a^4 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2}-\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {i a^4 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {a^3 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]

[Out]

1/2*a*(f*x+e)^2/b^2/f+2*a^2*(f*x+e)*arctan(exp(d*x+c))/b^3/d-2*(f*x+e)*arctan(exp(d*x+c))/b/d-2*a^4*(f*x+e)*ar
ctan(exp(d*x+c))/b^3/(a^2+b^2)/d-f*cosh(d*x+c)/b/d^2-a*(f*x+e)*ln(1+exp(2*d*x+2*c))/b^2/d+a^3*(f*x+e)*ln(1+exp
(2*d*x+2*c))/b^2/(a^2+b^2)/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d-a^3*(f*x+e)*ln
(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d+I*a^2*f*polylog(2,I*exp(d*x+c))/b^3/d^2-I*a^4*f*polylog(2
,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+I*a^4*f*polylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-I*f*polylog(2,I*exp(d*x+c)
)/b/d^2+I*f*polylog(2,-I*exp(d*x+c))/b/d^2-I*a^2*f*polylog(2,-I*exp(d*x+c))/b^3/d^2-1/2*a*f*polylog(2,-exp(2*d
*x+2*c))/b^2/d^2+1/2*a^3*f*polylog(2,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^2-a^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+
b^2)^(1/2)))/b^2/(a^2+b^2)/d^2-a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2+(f*x+e)*si
nh(d*x+c)/b/d

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5700, 5557, 3377, 2718, 4265, 2317, 2438, 3799, 2221, 5686, 5692, 5680, 6874} \begin {gather*} \frac {2 a^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^3 d}-\frac {i a^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {i a^2 f \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 a^4 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^3 d \left (a^2+b^2\right )}+\frac {i a^4 f \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac {i a^4 f \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2 \left (a^2+b^2\right )}-\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2 \left (a^2+b^2\right )}-\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {a^3 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2 \left (a^2+b^2\right )}-\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d \left (a^2+b^2\right )}+\frac {a^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d \left (a^2+b^2\right )}-\frac {a f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}-\frac {a (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d}+\frac {a (e+f x)^2}{2 b^2 f}-\frac {2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d}+\frac {i f \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac {i f \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {f \cosh (c+d x)}{b d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e + f*x)^2)/(2*b^2*f) + (2*a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)*ArcTan[E^(c + d*x)])/
(b*d) - (2*a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (f*Cosh[c + d*x])/(b*d^2) - (a^3*(e + f*x)
*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/
(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^2*d) + (a^3*(e + f*x)*
Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - (I*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) + (I*f*PolyLo
g[2, (-I)*E^(c + d*x)])/(b*d^2) + (I*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + (I*a^2*f*Poly
Log[2, I*E^(c + d*x)])/(b^3*d^2) - (I*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^4*f*PolyLog[2, I*E^(c + d*x)
])/(b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2)
- (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (a*f*PolyLog[2, -E^(2*(
c + d*x))])/(2*b^2*d^2) + (a^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^2) + ((e + f*x)*Sinh[c + d
*x])/(b*d)

Rule 2221

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

Rule 2317

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

Rule 2438

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

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, 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 5557

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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 6874

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

Rubi steps

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

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Mathematica [A]
time = 3.26, size = 481, normalized size = 0.76 \begin {gather*} -\frac {\frac {f \cosh (c+d x)}{b}+\frac {a^3 \left (-\frac {1}{2} f (c+d x)^2+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b^2 \left (a^2+b^2\right )}+\frac {-a d e (c+d x)+a c f (c+d x)+\frac {1}{2} a f (c+d x)^2+2 b d e \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))-2 b c f \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))+2 b f (c+d x) \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))+a f (c+d x) \log (2 \cosh (c+d x) (\cosh (c+d x)-\sinh (c+d x)))+a d e \log (1+\cosh (2 (c+d x))+\sinh (2 (c+d x)))-a c f \log (1+\cosh (2 (c+d x))+\sinh (2 (c+d x)))-i b f \text {PolyLog}(2,-i (\cosh (c+d x)+\sinh (c+d x)))+i b f \text {PolyLog}(2,i (\cosh (c+d x)+\sinh (c+d x)))-\frac {1}{2} a f \text {PolyLog}(2,-\cosh (2 (c+d x))+\sinh (2 (c+d x)))}{a^2+b^2}-\frac {d (e+f x) \sinh (c+d x)}{b}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((f*Cosh[c + d*x])/b + (a^3*(-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 + S
qrt[a^2 + b^2]))]))/(b^2*(a^2 + b^2)) + (-(a*d*e*(c + d*x)) + a*c*f*(c + d*x) + (a*f*(c + d*x)^2)/2 + 2*b*d*e*
ArcTan[Cosh[c + d*x] + Sinh[c + d*x]] - 2*b*c*f*ArcTan[Cosh[c + d*x] + Sinh[c + d*x]] + 2*b*f*(c + d*x)*ArcTan
[Cosh[c + d*x] + Sinh[c + d*x]] + a*f*(c + d*x)*Log[2*Cosh[c + d*x]*(Cosh[c + d*x] - Sinh[c + d*x])] + a*d*e*L
og[1 + Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]] - a*c*f*Log[1 + Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]] - I*b*f*P
olyLog[2, (-I)*(Cosh[c + d*x] + Sinh[c + d*x])] + I*b*f*PolyLog[2, I*(Cosh[c + d*x] + Sinh[c + d*x])] - (a*f*P
olyLog[2, -Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]])/2)/(a^2 + b^2) - (d*(e + f*x)*Sinh[c + d*x])/b)/d^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4065 vs. \(2 (592 ) = 1184\).
time = 5.85, size = 4066, normalized size = 6.44

method	result	size
risch	\(\text {Expression too large to display}\)	\(4066\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d^2*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d/b^2
*e*a^3/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2/b^2*a*f*c*ln(exp(d*x+c))-1/d^2/b^2*a^3*f/(a^2+b^2
)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2/b^2*a^3*f/(a^2+b^2)*dilog((b*exp(d*x+c)+
(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/2/d*a*f/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(
1/2)))*x-2/d^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))*a-4/d*b*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+2/d/b^2*e*a*ln
(exp(d*x+c))-1/2/d^2*a*f/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/2/d^2*a*f/(
a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/d*e/(2*a^2+2*b^2)*a*ln(b*exp(2*d*x+2*c)
+2*a*exp(d*x+c)-b)-2/d*e/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))-1/d^2*a^2*f/(a^2+b^2)^(3/2)*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*f/(a^2+b^2)^(3/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(
a+(a^2+b^2)^(1/2)))-1/d^2/b^2*a^4*f/(a^2+b^2)^(3/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2
)))+1/d^2/b^2*a^4*f/(a^2+b^2)^(3/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+4/d^2*b*f*c/(2
*a^2+2*b^2)*arctan(exp(d*x+c))+2/d/b^2*e*a^4/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))
-2/d*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*x-2/d^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*c-1/d*a^2*f/(a^2+b^2)^(
3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d^2*a^2*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c
)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/d^2*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a
^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/2/d^2*a*f*c/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/d^2/b^2
*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/d^2/b^2*a
^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/d^2/b^2*a^3*f
*c/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2/b^2*a^4*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))-1/d/b^2*a^3*f/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-
1/d^2/b^2*a^3*f/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d/b^2*a^3*f/(a^2+b^2)
*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d^2/b^2*a^3*f/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d/b^2*a^4*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b
^2)^(1/2)))*x+1/d^2/b^2*a^4*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d/b
^2*a^4*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d^2/b^2*a^4*f/(a^2+b^2
)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/d/b^2*a^4*e/(2*a^2+2*b^2)/(a^2+b^2)^(1/
2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d/b^2*a^2*e/(2*a^2+2*b^2)*(a^2+b^2)^(1/2)*arctanh(1/2*(
2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2*I/d*b*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-2*I/d^2*b*f/(2*a^2+2*b^2)*ln
(1-I*exp(d*x+c))*c+2*I/d*b*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x+2*I/d^2*b*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c
+1/2*a*f*x^2/b^2-2/d/b^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^
(1/2)))*x+2/d/b^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)
))*x-2/d^2*a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-2/d*
a^2*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+2/d^2*a^2*f/(2*
a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/d*a^2*f/(2*a^2+2*b^2
)/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d^2/b^2*f*a*c^2+2/d*e*a^2/(a^
2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d*e/(2*a^2+2*b^2)*(a^2+b^2)^(1/2)*arctanh(1/2
*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/d*e*a/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d^2*f/(2*a^
2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a+1/d^2*f/(2*a^2+2*b^2)*dilog((b*exp(d*
x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a-2/d^2*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))*a-1/2/d^2*a*f/(a^2+
b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/d^2*a*f*c/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))
-2/d^2*a^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))+2/d^2*f*c/(2*a^2+2*b^2)*(a^2+
b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d*a^2*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b
^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2*a^2*f/(a^2+b^2)^(3/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^
2)^(1/2)))*c-1/2/d*a*f/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/2/d^2*a*f/(a^2
+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+
b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a*x+1/d^2*f...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*f*(2*(a*d^2*x^2*e^c - (b*d*x*e^(2*c) - b*e^(2*c))*e^(d*x) + (b*d*x + b)*e^(-d*x))*e^(-c)/(b^2*d^2) - inte
grate(-8*(a^4*x*e^(d*x + c) - a^3*b*x)/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^3*b^2
*e^c + a*b^4*e^c)*e^(d*x)), x) + integrate(8*(b*x*e^(d*x + c) - a*x)/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*
e^(2*d*x)), x)) - 1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^4)*d) - 4*b*arctan(
e^(-d*x - c))/((a^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x
+ c)/(b*d) + e^(-d*x - c)/(b*d))*e

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1646 vs. \(2 (579) = 1158\).
time = 0.44, size = 1646, normalized size = 2.61 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b
+ b^3)*d*sinh(1) - (a^2*b + b^3)*f)*cosh(d*x + c)^2 + (a^2*b + b^3)*d*sinh(1) - ((a^2*b + b^3)*d*f*x + (a^2*b
+ b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^2*b + b^3)*f)*sinh(d*x + c)^2 + (a^2*b + b^3)*f - ((a^3 + a*b^
2)*d^2*f*x^2 - 2*(a^3 + a*b^2)*c^2*f + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*cosh(1) + 2*((a^3 + a*b^2
)*d^2*x + 2*(a^3 + a*b^2)*c*d)*sinh(1))*cosh(d*x + c) + 2*(a^3*f*cosh(d*x + c) + a^3*f*sinh(d*x + c))*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) + 2*(
a^3*f*cosh(d*x + c) + a^3*f*sinh(d*x + c))*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) + 2*((a*b^2*f + I*b^3*f)*cosh(d*x + c) + (a*b^2*f + I*b^3*f)*sin
h(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 2*((a*b^2*f - I*b^3*f)*cosh(d*x + c) + (a*b^2*f - I*b^3
*f)*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*((a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*co
sh(d*x + c) + (a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*((a^3*c*f - a^3*d*cosh(1) - a^3*d*sinh(1))*cosh(d*x + c) + (a^3*c*f
- a^3*d*cosh(1) - a^3*d*sinh(1))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^
2)/b^2) + 2*a) + 2*((a^3*d*f*x + a^3*c*f)*cosh(d*x + c) + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c))*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) + 2*((a^3*d*f*x +
a^3*c*f)*cosh(d*x + c) + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*co
sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*((a*b^2*c*f + I*b^3*c*f - a*b^2*d*cosh(1) - I
*b^3*d*cosh(1) - a*b^2*d*sinh(1) - I*b^3*d*sinh(1))*cosh(d*x + c) + (a*b^2*c*f + I*b^3*c*f - a*b^2*d*cosh(1) -
I*b^3*d*cosh(1) - a*b^2*d*sinh(1) - I*b^3*d*sinh(1))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) -
2*((a*b^2*c*f - I*b^3*c*f - a*b^2*d*cosh(1) + I*b^3*d*cosh(1) - a*b^2*d*sinh(1) + I*b^3*d*sinh(1))*cosh(d*x +
c) + (a*b^2*c*f - I*b^3*c*f - a*b^2*d*cosh(1) + I*b^3*d*cosh(1) - a*b^2*d*sinh(1) + I*b^3*d*sinh(1))*sinh(d*x
+ c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + 2*((a*b^2*d*f*x - I*b^3*d*f*x + a*b^2*c*f - I*b^3*c*f)*cosh(d*x
+ c) + (a*b^2*d*f*x - I*b^3*d*f*x + a*b^2*c*f - I*b^3*c*f)*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x +
c) + 1) + 2*((a*b^2*d*f*x + I*b^3*d*f*x + a*b^2*c*f + I*b^3*c*f)*cosh(d*x + c) + (a*b^2*d*f*x + I*b^3*d*f*x +
a*b^2*c*f + I*b^3*c*f)*sinh(d*x + c))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^3 + a*b^2)*d^2*f*x^2 -
2*(a^3 + a*b^2)*c^2*f + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*cosh(1) + 2*((a^2*b + b^3)*d*f*x + (a^2
*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^2*b + b^3)*f)*cosh(d*x + c) + 2*((a^3 + a*b^2)*d^2*x + 2*(a
^3 + a*b^2)*c*d)*sinh(1))*sinh(d*x + c))/((a^2*b^2 + b^4)*d^2*cosh(d*x + c) + (a^2*b^2 + b^4)*d^2*sinh(d*x + c
))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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