\(\int \sinh ^2(\frac {a+b x}{c+d x}) \, dx\) [296]

Optimal result

Integrand size = 16, antiderivative size = 107 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \] Output:

(-a*d+b*c)*Chi(2*(-a*d+b*c)/d/(d*x+c))*sinh(2*b/d)/d^2+(d*x+c)*sinh((b*x+a 
)/(d*x+c))^2/d-(-a*d+b*c)*cosh(2*b/d)*Shi(2*(-a*d+b*c)/d/(d*x+c))/d^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(475\) vs. \(2(107)=214\).

Time = 2.73 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.44 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 (a+b x)}{c+d x}}+c d e^{\frac {2 (a+b x)}{c+d x}}-2 d^2 x+2 d^2 x \cosh \left (\frac {2 b}{d}\right ) \cosh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 (b c-a d) \text {Chi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {2 b}{d}\right )-\sinh \left (\frac {2 b}{d}\right )\right )+2 (b c-a d) \text {Chi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right ) \left (\cosh \left (\frac {2 b}{d}\right )+\sinh \left (\frac {2 b}{d}\right )\right )+2 d^2 x \sinh \left (\frac {2 b}{d}\right ) \sinh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )}{4 d^2} \] Input:

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

Output:

((c*d)/E^((2*(a + b*x))/(c + d*x)) + c*d*E^((2*(a + b*x))/(c + d*x)) - 2*d 
^2*x + 2*d^2*x*Cosh[(2*b)/d]*Cosh[(2*(-(b*c) + a*d))/(d*(c + d*x))] - 2*(b 
*c - a*d)*CoshIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)]*(Cosh[(2*b)/d] - Sin 
h[(2*b)/d]) + 2*(b*c - a*d)*CoshIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] 
*(Cosh[(2*b)/d] + Sinh[(2*b)/d]) + 2*d^2*x*Sinh[(2*b)/d]*Sinh[(2*(-(b*c) + 
 a*d))/(d*(c + d*x))] + 2*b*c*Cosh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d) 
)/(d*(c + d*x))] - 2*a*d*Cosh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d* 
(c + d*x))] + 2*b*c*Sinh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d*(c + 
d*x))] - 2*a*d*Sinh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x)) 
] - 2*b*c*Cosh[(2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] + 2*a* 
d*Cosh[(2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] + 2*b*c*Sinh[( 
2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] - 2*a*d*Sinh[(2*b)/d]* 
SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)])/(4*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {6141, 3042, 25, 3794, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx\)

\(\Big \downarrow \) 6141

\(\displaystyle -\frac {\int (c+d x)^2 \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -(c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i (b c-a d)}{d (c+d x)}\right )^2d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i (b c-a d)}{d (c+d x)}\right )^2d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3794

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {2 i (b c-a d) \int \frac {1}{2} i (c+d x) \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {(b c-a d) \int (c+d x) \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}-\left ((c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\left ((c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )-\frac {(b c-a d) \int -i (c+d x) \sin \left (\frac {2 i b}{d}-\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \int (c+d x) \sin \left (\frac {2 i b}{d}-\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \cosh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}+\cosh \left (\frac {2 b}{d}\right ) \int -i (c+d x) \sinh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \cosh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \int (c+d x) \sinh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \int -i (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-\cosh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )\right )}{d}}{d}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {-(c+d x) \sinh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )-i \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )\right )}{d}}{d}\)

Input:

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

Output:

-((-((c + d*x)*Sinh[b/d - (b*c - a*d)/(d*(c + d*x))]^2) + (I*(b*c - a*d)*( 
I*CoshIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sinh[(2*b)/d] - I*Cosh[(2*b) 
/d]*SinhIntegral[(2*(b*c - a*d))/(d*(c + d*x))]))/d)/d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

rule 6141

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol 
] :> Simp[-d^(-1)   Subst[Int[Sinh[b*(e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x] 
, x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - 
a*d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(107)=214\).

Time = 2.61 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.35

method	result	size
risch	\(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} a}{\frac {4 d a}{d x +c}-\frac {4 b c}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} b c}{4 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (\frac {2 d a -2 b c}{\left (d x +c \right ) d}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (\frac {2 d a -2 b c}{\left (d x +c \right ) d}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x a}{4 d a -4 b c}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x b c}{4 \left (d a -b c \right )}+\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c a}{4 d a -4 b c}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c^{2} b}{4 d \left (d a -b c \right )}+\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {2 \left (d a -b c \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {2 \left (d a -b c \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\)	\(358\)

Input:

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

Output:

-1/2*x+1/4*exp(-2*(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c))*a-1/4/d*exp(- 
2*(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c))*b*c-1/2/d*exp(-2*b/d)*Ei(1,2* 
(a*d-b*c)/d/(d*x+c))*a+1/2/d^2*exp(-2*b/d)*Ei(1,2*(a*d-b*c)/d/(d*x+c))*b*c 
+1/4*d*exp(2*(b*x+a)/(d*x+c))/(a*d-b*c)*x*a-1/4*exp(2*(b*x+a)/(d*x+c))/(a* 
d-b*c)*x*b*c+1/4*exp(2*(b*x+a)/(d*x+c))/(a*d-b*c)*c*a-1/4/d*exp(2*(b*x+a)/ 
(d*x+c))/(a*d-b*c)*c^2*b+1/2/d*exp(2*b/d)*Ei(1,-2*(a*d-b*c)/d/(d*x+c))*a-1 
/2/d^2*exp(2*b/d)*Ei(1,-2*(a*d-b*c)/d/(d*x+c))*b*c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (107) = 214\).

Time = 0.11 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.46 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (d^{2} x - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2}\right )}} \] Input:

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

Output:

-1/2*(d^2*x - (d^2*x + c*d)*cosh((b*x + a)/(d*x + c))^2 - (d^2*x - (b*c - 
a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh(2*b/d) + c*d)*sinh((b*x + a)/(d 
*x + c))^2 - ((b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/ 
(d*x + c))^2 - (b*c - a*d)*Ei(2*(b*c - a*d)/(d^2*x + c*d)))*cosh(2*b/d) - 
((b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2 
- (b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*sinh((b*x + a)/(d*x + c))^2 
 + (b*c - a*d)*Ei(2*(b*c - a*d)/(d^2*x + c*d)))*sinh(2*b/d))/(d^2*cosh((b* 
x + a)/(d*x + c))^2 - d^2*sinh((b*x + a)/(d*x + c))^2)
 

Sympy [F(-1)]

Timed out. \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x + a}{d x + c}\right )^{2} \,d x } \] Input:

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

Output:

-1/2*x + 1/4*integrate(e^(2*b*c/(d^2*x + c*d) - 2*a/(d*x + c) - 2*b/d), x) 
 + 1/4*integrate(e^(-2*b*c/(d^2*x + c*d) + 2*a/(d*x + c) + 2*b/d), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (107) = 214\).

Time = 7.90 (sec) , antiderivative size = 749, normalized size of antiderivative = 7.00 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(2*b^3*c^2*Ei(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) - 4*a*b^2*c* 
d*Ei(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) - 2*(b*x + a)*b^2*c^2*d*E 
i(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) + 2*a^2*b*d^2*Ei(- 
2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) + 4*(b*x + a)*a*b*c*d^2*Ei(-2*( 
b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) - 2*(b*x + a)*a^2*d^3*Ei 
(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) - 2*b^3*c^2*Ei(2*(b 
 - (b*x + a)*d/(d*x + c))/d)*e^(-2*b/d) + 4*a*b^2*c*d*Ei(2*(b - (b*x + a)* 
d/(d*x + c))/d)*e^(-2*b/d) + 2*(b*x + a)*b^2*c^2*d*Ei(2*(b - (b*x + a)*d/( 
d*x + c))/d)*e^(-2*b/d)/(d*x + c) - 2*a^2*b*d^2*Ei(2*(b - (b*x + a)*d/(d*x 
 + c))/d)*e^(-2*b/d) - 4*(b*x + a)*a*b*c*d^2*Ei(2*(b - (b*x + a)*d/(d*x + 
c))/d)*e^(-2*b/d)/(d*x + c) + 2*(b*x + a)*a^2*d^3*Ei(2*(b - (b*x + a)*d/(d 
*x + c))/d)*e^(-2*b/d)/(d*x + c) + b^2*c^2*d*e^(2*(b*x + a)/(d*x + c)) - 2 
*a*b*c*d^2*e^(2*(b*x + a)/(d*x + c)) + a^2*d^3*e^(2*(b*x + a)/(d*x + c)) + 
 b^2*c^2*d*e^(-2*(b*x + a)/(d*x + c)) - 2*a*b*c*d^2*e^(-2*(b*x + a)/(d*x + 
 c)) + a^2*d^3*e^(-2*(b*x + a)/(d*x + c)) - 2*b^2*c^2*d + 4*a*b*c*d^2 - 2* 
a^2*d^3)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d 
*x + c))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(2*e**((4*a + 4*b*x)/(c + d*x))*a*d**2*x**2 - 2*e**((4*a + 4*b*x)/(c + d*x 
))*b*c*d*x**2 - e**((4*a + 4*b*x)/(c + d*x))*c**3 - e**((4*a + 4*b*x)/(c + 
 d*x))*c**2*d*x - 4*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b* 
x)/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a 
+ 2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3), 
x)*a**2*c*d**3 - 4*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b*x 
)/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a + 
 2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3),x 
)*a**2*d**4*x + 8*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b*x) 
/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a + 
2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3),x) 
*a*b*c**2*d**2 + 8*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b*x 
)/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a + 
 2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3),x 
)*a*b*c*d**3*x - 4*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b*x 
)/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a + 
 2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3),x 
)*b**2*c**3*d - 4*e**((2*a + 2*b*x)/(c + d*x))*int(x**2/(e**((2*a + 2*b*x) 
/(c + d*x))*c**3 + 3*e**((2*a + 2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*a + 
2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*a + 2*b*x)/(c + d*x))*d**3*x**3)...