\(\int \sinh ^3(e+\frac {f (a+b x)}{c+d x}) \, dx\) [300]

Optimal result

Integrand size = 19, antiderivative size = 226 \[ \int \sinh ^3\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=-\frac {3 (b c-a d) f \cosh \left (e+\frac {b f}{d}\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac {3 (b c-a d) f \cosh \left (3 \left (e+\frac {b f}{d}\right )\right ) \text {Chi}\left (\frac {3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac {3 (b c-a d) f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) f \sinh \left (3 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2} \] Output:

-3/4*(-a*d+b*c)*f*cosh(e+b*f/d)*Chi((-a*d+b*c)*f/d/(d*x+c))/d^2+3/4*(-a*d+ 
b*c)*f*cosh(3*e+3*b*f/d)*Chi(3*(-a*d+b*c)*f/d/(d*x+c))/d^2+(d*x+c)*sinh((b 
*f*x+d*e*x+a*f+c*e)/(d*x+c))^3/d+3/4*(-a*d+b*c)*f*sinh(e+b*f/d)*Shi((-a*d+ 
b*c)*f/d/(d*x+c))/d^2-3/4*(-a*d+b*c)*f*sinh(3*e+3*b*f/d)*Shi(3*(-a*d+b*c)* 
f/d/(d*x+c))/d^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(913\) vs. \(2(226)=452\).

Time = 6.46 (sec) , antiderivative size = 913, normalized size of antiderivative = 4.04 \[ \int \sinh ^3\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*c/(d*E^((3*(c*e + a*f + d*e*x + b*f*x))/(c + d*x))) + (3*c)/(8*d*E^(( 
c*e + a*f + d*e*x + b*f*x)/(c + d*x))) - (3*c*E^((c*e + a*f + d*e*x + b*f* 
x)/(c + d*x)))/(8*d) + (c*E^((3*(c*e + a*f + d*e*x + b*f*x))/(c + d*x)))/( 
8*d) - (3*x*Cosh[(-(b*c*f) + a*d*f)/(d*(c + d*x))]*Sinh[(d*e + b*f)/d])/4 
+ (x*Cosh[(3*(-(b*c*f) + a*d*f))/(d*(c + d*x))]*Sinh[(3*(d*e + b*f))/d])/4 
 - (3*x*Cosh[(d*e + b*f)/d]*Sinh[(-(b*c*f) + a*d*f)/(d*(c + d*x))])/4 + (x 
*Cosh[(3*(d*e + b*f))/d]*Sinh[(3*(-(b*c*f) + a*d*f))/(d*(c + d*x))])/4 - ( 
3*(-(b*c) + a*d)*f*(Cosh[(3*(d*e + b*f))/d]*CoshIntegral[(3*b*c*f - 3*a*d* 
f)/(c*d + d^2*x)] - Cosh[(d*e + b*f)/d]*CoshIntegral[(b*c*f - a*d*f)/(c*d 
+ d^2*x)] - Cosh[(d*e + b*f)/d]*CoshIntegral[(-(b*c*f) + a*d*f)/(c*d + d^2 
*x)] + Cosh[(3*(d*e + b*f))/d]*CoshIntegral[(-3*b*c*f + 3*a*d*f)/(c*d + d^ 
2*x)] + CoshIntegral[(b*c*f - a*d*f)/(c*d + d^2*x)]*Sinh[(d*e + b*f)/d] - 
CoshIntegral[(-(b*c*f) + a*d*f)/(c*d + d^2*x)]*Sinh[(d*e + b*f)/d] - CoshI 
ntegral[(3*b*c*f - 3*a*d*f)/(c*d + d^2*x)]*Sinh[(3*(d*e + b*f))/d] + CoshI 
ntegral[(-3*b*c*f + 3*a*d*f)/(c*d + d^2*x)]*Sinh[(3*(d*e + b*f))/d] + Cosh 
[(3*(d*e + b*f))/d]*SinhIntegral[(3*b*c*f - 3*a*d*f)/(c*d + d^2*x)] - Sinh 
[(3*(d*e + b*f))/d]*SinhIntegral[(3*b*c*f - 3*a*d*f)/(c*d + d^2*x)] - Cosh 
[(d*e + b*f)/d]*SinhIntegral[(b*c*f - a*d*f)/(c*d + d^2*x)] + Sinh[(d*e + 
b*f)/d]*SinhIntegral[(b*c*f - a*d*f)/(c*d + d^2*x)] - Cosh[(d*e + b*f)/d]* 
SinhIntegral[(-(b*c*f) + a*d*f)/(c*d + d^2*x)] - Sinh[(d*e + b*f)/d]*Si...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6143, 6141, 3042, 26, 3794, 2009}

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.

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

\(\Big \downarrow \) 6143

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

\(\Big \downarrow \) 6141

\(\displaystyle -\frac {\int (c+d x)^2 \sinh ^3\left (e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int i (c+d x)^2 \sin \left (i \left (e+\frac {b f}{d}\right )-\frac {i (b c-a d) f}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (c+d x)^2 \sin \left (i \left (e+\frac {b f}{d}\right )-\frac {i (b c-a d) f}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3794

\(\displaystyle -\frac {i \left (i (c+d x) \sinh ^3\left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )-\frac {3 i f (b c-a d) \int \left (\frac {1}{4} (c+d x) \cosh \left (e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}\right )-\frac {1}{4} (c+d x) \cosh \left (3 \left (e+\frac {b f}{d}\right )-\frac {3 (b c-a d) f}{d (c+d x)}\right )\right )d\frac {1}{c+d x}}{d}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \left (i (c+d x) \sinh ^3\left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )-\frac {3 i f (b c-a d) \left (\frac {1}{4} \cosh \left (\frac {b f}{d}+e\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-\frac {1}{4} \cosh \left (3 \left (\frac {b f}{d}+e\right )\right ) \text {Chi}\left (\frac {3 (b c-a d) f}{d (c+d x)}\right )-\frac {1}{4} \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+\frac {1}{4} \sinh \left (3 \left (\frac {b f}{d}+e\right )\right ) \text {Shi}\left (\frac {3 (b c-a d) f}{d (c+d x)}\right )\right )}{d}\right )}{d}\)

Input:

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

Output:

((-I)*(I*(c + d*x)*Sinh[e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))]^3 - ( 
(3*I)*(b*c - a*d)*f*((Cosh[e + (b*f)/d]*CoshIntegral[((b*c - a*d)*f)/(d*(c 
 + d*x))])/4 - (Cosh[3*(e + (b*f)/d)]*CoshIntegral[(3*(b*c - a*d)*f)/(d*(c 
 + d*x))])/4 - (Sinh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x 
))])/4 + (Sinh[3*(e + (b*f)/d)]*SinhIntegral[(3*(b*c - a*d)*f)/(d*(c + d*x 
))])/4))/d))/d
 

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

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. \(929\) vs. \(2(220)=440\).

Time = 5.83 (sec) , antiderivative size = 930, normalized size of antiderivative = 4.12

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {3 \left (b f x +d e x +a f +c e \right )}{d x +c}} a f}{8 \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {3 \left (b f x +d e x +a f +c e \right )}{d x +c}} b c f}{8 d \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 \left (b f +d e \right )}{d}} \operatorname {expIntegral}_{1}\left (\frac {3 d a f -3 b c f}{d \left (d x +c \right )}\right ) a f}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {3 \left (b f +d e \right )}{d}} \operatorname {expIntegral}_{1}\left (\frac {3 d a f -3 b c f}{d \left (d x +c \right )}\right ) b c f}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} a f}{8 \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} b c f}{8 d \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b f +d e}{d}} \operatorname {expIntegral}_{1}\left (\frac {d a f -b c f}{d \left (d x +c \right )}\right ) a f}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {b f +d e}{d}} \operatorname {expIntegral}_{1}\left (\frac {d a f -b c f}{d \left (d x +c \right )}\right ) b c f}{8 d^{2}}+\frac {{\mathrm e}^{\frac {3 b f x +3 d e x +3 a f +3 c e}{d x +c}} a f}{8 d \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}-\frac {{\mathrm e}^{\frac {3 b f x +3 d e x +3 a f +3 c e}{d x +c}} b c f}{8 d^{2} \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}+\frac {3 \,{\mathrm e}^{\frac {3 b f +3 d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {3 \left (d a f -b c f \right )}{d \left (d x +c \right )}-\frac {3 \left (b f +d e \right )}{d}-\frac {3 \left (-b f -d e \right )}{d}\right ) a f}{8 d}-\frac {3 \,{\mathrm e}^{\frac {3 b f +3 d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {3 \left (d a f -b c f \right )}{d \left (d x +c \right )}-\frac {3 \left (b f +d e \right )}{d}-\frac {3 \left (-b f -d e \right )}{d}\right ) b c f}{8 d^{2}}-\frac {3 \,{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} a f}{8 d \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}+\frac {3 \,{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} b c f}{8 d^{2} \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}-\frac {3 \,{\mathrm e}^{\frac {b f +d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {d a f -b c f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) a f}{8 d}+\frac {3 \,{\mathrm e}^{\frac {b f +d e}{d}} \operatorname {expIntegral}_{1}\left (-\frac {d a f -b c f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) b c f}{8 d^{2}}\)	\(930\)

Input:

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

Output:

-1/8*exp(-3*(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+1/8/d*exp(-3*(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+3/8/d*exp(-3*(b*f+d*e)/d)*Ei(1,3/d*(a*d*f-b*c*f)/(d*x+c))*a*f 
-3/8/d^2*exp(-3*(b*f+d*e)/d)*Ei(1,3/d*(a*d*f-b*c*f)/(d*x+c))*b*c*f+3/8*exp 
(-(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-3/8/d 
*exp(-(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 
-3/8/d*exp(-(b*f+d*e)/d)*Ei(1,1/d*(a*d*f-b*c*f)/(d*x+c))*a*f+3/8/d^2*exp(- 
(b*f+d*e)/d)*Ei(1,1/d*(a*d*f-b*c*f)/(d*x+c))*b*c*f+1/8/d*exp(3*(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-1/8/d^2*exp(3*(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+3/8/d*exp 
(3*(b*f+d*e)/d)*Ei(1,-3/d*(a*d*f-b*c*f)/(d*x+c)-3*(b*f+d*e)/d-3*(-b*f-d*e) 
/d)*a*f-3/8/d^2*exp(3*(b*f+d*e)/d)*Ei(1,-3/d*(a*d*f-b*c*f)/(d*x+c)-3*(b*f+ 
d*e)/d-3*(-b*f-d*e)/d)*b*c*f-3/8/d*exp((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+3/8/d^2*exp((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-3/8/d*exp((b*f+d*e)/d)*Ei(1,-1/d*(a 
*d*f-b*c*f)/(d*x+c)-(b*f+d*e)/d-(-b*f-d*e)/d)*a*f+3/8/d^2*exp((b*f+d*e)/d) 
*Ei(1,-1/d*(a*d*f-b*c*f)/(d*x+c)-(b*f+d*e)/d-(-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. 942 vs. \(2 (220) = 440\).

Time = 0.13 (sec) , antiderivative size = 942, normalized size of antiderivative = 4.17 \[ \int \sinh ^3\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*(6*(b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + 
 (d*e + b*f)*x)/(d*x + c))^2*cosh(3*(d*e + b*f)/d)*sinh((c*e + a*f + (d*e 
+ b*f)*x)/(d*x + c))^2 - 3*(b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d) 
)*cosh(3*(d*e + b*f)/d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4 - 2* 
(d^2*x + c*d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^3 - 3*((b*c - a* 
d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/( 
d*x + c))^4 + (b*c - a*d)*f*Ei(3*(b*c - a*d)*f/(d^2*x + c*d)))*cosh(3*(d*e 
 + b*f)/d) + 3*((b*c - a*d)*f*Ei((b*c - a*d)*f/(d^2*x + c*d)) + (b*c - a*d 
)*f*Ei(-(b*c - a*d)*f/(d^2*x + c*d)))*cosh((d*e + b*f)/d) + 6*(d^2*x - (d^ 
2*x + c*d)*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 + c*d)*sinh((c*e 
+ a*f + (d*e + b*f)*x)/(d*x + c)) - 3*((b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/( 
d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4 - 2*(b*c - a*d 
)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d 
*x + c))^2*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 + (b*c - a*d)*f*E 
i(-3*(b*c - a*d)*f/(d^2*x + c*d))*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + 
c))^4 - (b*c - a*d)*f*Ei(3*(b*c - a*d)*f/(d^2*x + c*d)))*sinh(3*(d*e + b*f 
)/d) - 3*((b*c - a*d)*f*Ei((b*c - a*d)*f/(d^2*x + c*d)) - (b*c - a*d)*f*Ei 
(-(b*c - a*d)*f/(d^2*x + c*d)))*sinh((d*e + b*f)/d))/(d^2*cosh((c*e + a*f 
+ (d*e + b*f)*x)/(d*x + c))^4 - 2*d^2*cosh((c*e + a*f + (d*e + b*f)*x)/(d* 
x + c))^2*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 + d^2*sinh((c*e...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3021 vs. \(2 (220) = 440\).

Time = 27.91 (sec) , antiderivative size = 3021, normalized size of antiderivative = 13.37 \[ \int \sinh ^3\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(3*b^2*c^2*d*e*f^2*Ei(-3*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d 
*x + c))/d)*e^(3*(d*e + b*f)/d) - 6*a*b*c*d^2*e*f^2*Ei(-3*(d*e + b*f - (d* 
e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(3*(d*e + b*f)/d) + 3*a^2*d^3*e 
*f^2*Ei(-3*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(3*( 
d*e + b*f)/d) + 3*b^3*c^2*f^3*Ei(-3*(d*e + b*f - (d*e*x + b*f*x + c*e + a* 
f)*d/(d*x + c))/d)*e^(3*(d*e + b*f)/d) - 6*a*b^2*c*d*f^3*Ei(-3*(d*e + b*f 
- (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(3*(d*e + b*f)/d) + 3*a^2* 
b*d^2*f^3*Ei(-3*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e 
^(3*(d*e + b*f)/d) - 3*b^2*c^2*d*e*f^2*Ei(-(d*e + b*f - (d*e*x + b*f*x + c 
*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) + 6*a*b*c*d^2*e*f^2*Ei(-(d*e + 
 b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) - 3*a 
^2*d^3*e*f^2*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)* 
e^((d*e + b*f)/d) - 3*b^3*c^2*f^3*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + 
a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) + 6*a*b^2*c*d*f^3*Ei(-(d*e + b*f - 
(d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) - 3*a^2*b*d^ 
2*f^3*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e 
 + b*f)/d) - 3*b^2*c^2*d*e*f^2*Ei((d*e + b*f - (d*e*x + b*f*x + c*e + a*f) 
*d/(d*x + c))/d)*e^(-(d*e + b*f)/d) + 6*a*b*c*d^2*e*f^2*Ei((d*e + b*f - (d 
*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-(d*e + b*f)/d) - 3*a^2*d^3*e 
*f^2*Ei((d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-(d...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sinh ^3\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {too large to display} \] Input:

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

Output:

(3*e**((6*a*f + 6*b*f*x + 6*c*e + 6*d*e*x)/(c + d*x))*a*d**2*f*x**2 - 3*e* 
*((6*a*f + 6*b*f*x + 6*c*e + 6*d*e*x)/(c + d*x))*b*c*d*f*x**2 - e**((6*a*f 
 + 6*b*f*x + 6*c*e + 6*d*e*x)/(c + d*x))*c**3 - e**((6*a*f + 6*b*f*x + 6*c 
*e + 6*d*e*x)/(c + d*x))*c**2*d*x - 9*e**((4*a*f + 4*b*f*x + 4*c*e + 4*d*e 
*x)/(c + d*x))*a*d**2*f*x**2 + 9*e**((4*a*f + 4*b*f*x + 4*c*e + 4*d*e*x)/( 
c + d*x))*b*c*d*f*x**2 + 9*e**((4*a*f + 4*b*f*x + 4*c*e + 4*d*e*x)/(c + d* 
x))*c**3 + 9*e**((4*a*f + 4*b*f*x + 4*c*e + 4*d*e*x)/(c + d*x))*c**2*d*x - 
 9*e**((3*a*f + 3*b*f*x + 3*c*e + 3*d*e*x)/(c + d*x))*int(x**2/(e**((a*f + 
 b*f*x + c*e + d*e*x)/(c + d*x))*c**3 + 3*e**((a*f + b*f*x + c*e + d*e*x)/ 
(c + d*x))*c**2*d*x + 3*e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*c*d**2* 
x**2 + e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*d**3*x**3),x)*a**2*c*d** 
3*f**2 - 9*e**((3*a*f + 3*b*f*x + 3*c*e + 3*d*e*x)/(c + d*x))*int(x**2/(e* 
*((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*c**3 + 3*e**((a*f + b*f*x + c*e + 
 d*e*x)/(c + d*x))*c**2*d*x + 3*e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x)) 
*c*d**2*x**2 + e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*d**3*x**3),x)*a* 
*2*d**4*f**2*x + 18*e**((3*a*f + 3*b*f*x + 3*c*e + 3*d*e*x)/(c + d*x))*int 
(x**2/(e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*c**3 + 3*e**((a*f + b*f* 
x + c*e + d*e*x)/(c + d*x))*c**2*d*x + 3*e**((a*f + b*f*x + c*e + d*e*x)/( 
c + d*x))*c*d**2*x**2 + e**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*d**3*x* 
*3),x)*a*b*c**2*d**2*f**2 + 18*e**((3*a*f + 3*b*f*x + 3*c*e + 3*d*e*x)/...