\(\int \sinh ^2(e+\frac {f (a+b x)}{c+d x}) \, dx\) [299]

Optimal result

Integrand size = 19, antiderivative size = 129 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(572\) vs. \(2(129)=258\).

Time = 3.35 (sec) , antiderivative size = 572, normalized size of antiderivative = 4.43 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 (c e+a f+d e x+b f x)}{c+d x}}+c d e^{\frac {2 (c e+a f+d e x+b f x)}{c+d x}}+2 d^2 x \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \cosh \left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+2 d^2 x \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \sinh \left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )-2 \left (d^2 x+(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \left (\cosh \left (2 \left (e+\frac {b f}{d}\right )\right )-\sinh \left (2 \left (e+\frac {b f}{d}\right )\right )\right )-(b c-a d) f \text {Chi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right ) \left (\cosh \left (2 \left (e+\frac {b f}{d}\right )\right )+\sinh \left (2 \left (e+\frac {b f}{d}\right )\right )\right )+b c f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-a d f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-b c f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )+a d f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-b c f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+a d f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )-b c f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+a d f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )\right )}{4 d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {6143, 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.

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

\(\Big \downarrow \) 6143

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

\(\Big \downarrow \) 6141

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

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

Output:

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

rule 6143

Int[Sinh[u_]^(n_.), x_Symbol] :> With[{lst = QuotientOfLinearsParts[u, x]}, 
 Int[Sinh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n 
, 0] && QuotientOfLinearsQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(471\) vs. \(2(131)=262\).

Time = 3.48 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.66

method	result	size
risch	\(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b f x +d e x +a f +c e \right )}{d x +c}} a f}{\frac {4 d f a}{d x +c}-\frac {4 b c f}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b f x +d e x +a f +c e \right )}{d x +c}} b c f}{4 d \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 \left (b f +d e \right )}{d}} \operatorname {expIntegral}_{1}\left (\frac {2 d a f -2 b c f}{d \left (d x +c \right )}\right ) a f}{2 d}+\frac {{\mathrm e}^{-\frac {2 \left (b f +d e \right )}{d}} \operatorname {expIntegral}_{1}\left (\frac {2 d a f -2 b c f}{d \left (d x +c \right )}\right ) b c f}{2 d^{2}}+\frac {{\mathrm e}^{\frac {2 b f x +2 d e x +2 a f +2 c e}{d x +c}} a f}{4 d \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}-\frac {{\mathrm e}^{\frac {2 b f x +2 d e x +2 a f +2 c e}{d x +c}} b c f}{4 d^{2} \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}+\frac {{\mathrm e}^{\frac {2 b f +2 d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {2 \left (d a f -b c f \right )}{d \left (d x +c \right )}-\frac {2 \left (b f +d e \right )}{d}-\frac {2 \left (-b f -d e \right )}{d}\right ) a f}{2 d}-\frac {{\mathrm e}^{\frac {2 b f +2 d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {2 \left (d a f -b c f \right )}{d \left (d x +c \right )}-\frac {2 \left (b f +d e \right )}{d}-\frac {2 \left (-b f -d e \right )}{d}\right ) b c f}{2 d^{2}}\)	\(472\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (131) = 262\).

Time = 0.09 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.70 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=-\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} + {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} + {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1596 vs. \(2 (131) = 262\).

Time = 22.72 (sec) , antiderivative size = 1596, normalized size of antiderivative = 12.37 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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