\(\int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx\) [22]

Optimal result

Integrand size = 16, antiderivative size = 184 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 b^2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{8 d^3}-\frac {3 b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{8 d^3}-\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \] Output:

9/8*b^2*Chi(3*b*c/d+3*b*x)*sinh(3*a-3*b*c/d)/d^3-3/8*b^2*Chi(b*c/d+b*x)*si 
nh(a-b*c/d)/d^3-3/2*b*cosh(b*x+a)*sinh(b*x+a)^2/d^2/(d*x+c)-1/2*sinh(b*x+a 
)^3/d/(d*x+c)^2-3/8*b^2*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^3+9/8*b^2*cosh(3*a- 
3*b*c/d)*Shi(3*b*c/d+3*b*x)/d^3
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.20 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {6 d \cosh (b x) (b (c+d x) \cosh (a)+d \sinh (a))-2 d \cosh (3 b x) (3 b (c+d x) \cosh (3 a)+d \sinh (3 a))+6 d (d \cosh (a)+b (c+d x) \sinh (a)) \sinh (b x)-2 d (d \cosh (3 a)+3 b (c+d x) \sinh (3 a)) \sinh (3 b x)+6 b^2 (c+d x)^2 \left (3 \text {Chi}\left (\frac {3 b (c+d x)}{d}\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-\text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )}{16 d^3 (c+d x)^2} \] Input:

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

Output:

(6*d*Cosh[b*x]*(b*(c + d*x)*Cosh[a] + d*Sinh[a]) - 2*d*Cosh[3*b*x]*(3*b*(c 
 + d*x)*Cosh[3*a] + d*Sinh[3*a]) + 6*d*(d*Cosh[a] + b*(c + d*x)*Sinh[a])*S 
inh[b*x] - 2*d*(d*Cosh[3*a] + 3*b*(c + d*x)*Sinh[3*a])*Sinh[3*b*x] + 6*b^2 
*(c + d*x)^2*(3*CoshIntegral[(3*b*(c + d*x))/d]*Sinh[3*a - (3*b*c)/d] - Co 
shIntegral[b*(c/d + x)]*Sinh[a - (b*c)/d] - Cosh[a - (b*c)/d]*SinhIntegral 
[b*(c/d + x)] + 3*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x))/d]))/ 
(16*d^3*(c + d*x)^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.43, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 26, 3795, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782, 3793, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i a+i b x)^3}{(c+d x)^3}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i a+i b x)^3}{(c+d x)^3}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle i \left (\frac {9 b^2 \int -\frac {i \sinh ^3(a+b x)}{c+d x}dx}{2 d^2}-\frac {3 b^2 \int \frac {i \sinh (a+b x)}{c+d x}dx}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {9 i b^2 \int \frac {\sinh ^3(a+b x)}{c+d x}dx}{2 d^2}-\frac {3 i b^2 \int \frac {\sinh (a+b x)}{c+d x}dx}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (-\frac {3 i b^2 \int -\frac {i \sin (i a+i b x)}{c+d x}dx}{d^2}-\frac {9 i b^2 \int \frac {i \sin (i a+i b x)^3}{c+d x}dx}{2 d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {3 b^2 \int \frac {\sin (i a+i b x)}{c+d x}dx}{d^2}+\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3784

\(\displaystyle i \left (\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {i \sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+i \cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (a-\frac {b c}{d}\right ) \int -\frac {i \sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle i \left (-\frac {3 b^2 \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}+\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3782

\(\displaystyle i \left (\frac {9 b^2 \int \frac {\sin (i a+i b x)^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle i \left (\frac {9 b^2 \int \left (\frac {3 i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right )dx}{2 d^2}-\frac {3 b^2 \left (\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (-\frac {3 b^2 \left (\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}+\frac {9 b^2 \left (-\frac {i \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}+\frac {3 i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {3 i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {i \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\right )}{2 d^2}+\frac {3 i b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {i \sinh ^3(a+b x)}{2 d (c+d x)^2}\right )\)

Input:

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

Output:

I*((((3*I)/2)*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(d^2*(c + d*x)) + ((I/2)*Si 
nh[a + b*x]^3)/(d*(c + d*x)^2) - (3*b^2*((I*CoshIntegral[(b*c)/d + b*x]*Si 
nh[a - (b*c)/d])/d + (I*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d)) 
/d^2 + (9*b^2*(((-1/4*I)*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c 
)/d])/d + (((3*I)/4)*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/d + (( 
(3*I)/4)*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d - ((I/4)*Cosh[3* 
a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/d))/(2*d^2))
 

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 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 3793

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

rule 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(561\) vs. \(2(172)=344\).

Time = 0.93 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.05

method	result	size
risch	\(-\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} x}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} c}{16 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-3 b x -3 a}}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {9 b^{2} {\mathrm e}^{-\frac {3 \left (d a -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (3 b x +3 a -\frac {3 \left (d a -b c \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{3} {\mathrm e}^{-b x -a} x}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-b x -a} c}{16 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-b x -a}}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{16 d^{3}}+\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}+\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}+\frac {3 b^{2} {\mathrm e}^{\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-d a +b c}{d}\right )}{16 d^{3}}-\frac {b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {9 b^{2} {\mathrm e}^{\frac {3 d a -3 b c}{d}} \operatorname {expIntegral}_{1}\left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right )}{16 d^{3}}\)	\(562\)

Input:

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

Output:

-3/16*b^3*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x-3/16*b^3*e 
xp(-3*b*x-3*a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c+1/16*b^2*exp(-3*b*x 
-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)+9/16*b^2/d^3*exp(-3*(a*d-b*c)/d) 
*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)+3/16*b^3*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c 
*d*x+b^2*c^2)*x+3/16*b^3*exp(-b*x-a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2) 
*c-3/16*b^2*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-3/16*b^2/d^3*e 
xp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b* 
x)^2+3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b*x)+3/16*b^2/d^3*exp((a*d-b*c)/d)*Ei( 
1,-b*x-a-(-a*d+b*c)/d)-1/16*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)^2-3/16*b^2/ 
d^3*exp(3*b*x+3*a)/(b*c/d+b*x)-9/16*b^2/d^3*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x 
-3*a-3*(-a*d+b*c)/d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (172) = 344\).

Time = 0.09 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.88 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {2 \, d^{2} \sinh \left (b x + a\right )^{3} + 6 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{3} + 18 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) + 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (d^{2} \cosh \left (b x + a\right )^{2} - d^{2}\right )} \sinh \left (b x + a\right ) + 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

-1/16*(2*d^2*sinh(b*x + a)^3 + 6*(b*d^2*x + b*c*d)*cosh(b*x + a)^3 + 18*(b 
*d^2*x + b*c*d)*cosh(b*x + a)*sinh(b*x + a)^2 - 6*(b*d^2*x + b*c*d)*cosh(b 
*x + a) + 3*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) - ( 
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a* 
d)/d) - 9*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) - ( 
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c 
- a*d)/d) + 6*(d^2*cosh(b*x + a)^2 - d^2)*sinh(b*x + a) + 3*((b^2*d^2*x^2 
+ 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x 
+ b^2*c^2)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 9*((b^2*d^2*x^2 + 
2*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x 
+ b^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^5*x^2 + 2*c* 
d^4*x + c^2*d^3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.79 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{3}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{3}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} \] Input:

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

Output:

1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(3, 3*(d*x + c)*b/d)/((d*x + c)^2*d) 
- 3/8*e^(-a + b*c/d)*exp_integral_e(3, (d*x + c)*b/d)/((d*x + c)^2*d) + 3/ 
8*e^(a - b*c/d)*exp_integral_e(3, -(d*x + c)*b/d)/((d*x + c)^2*d) - 1/8*e^ 
(3*a - 3*b*c/d)*exp_integral_e(3, -3*(d*x + c)*b/d)/((d*x + c)^2*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (172) = 344\).

Time = 0.20 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.27 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 6 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 18 \, b^{2} c d x {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 9 \, b^{2} c^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 9 \, b^{2} c^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} - 3 \, b d^{2} x e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b d^{2} x e^{\left (b x + a\right )} + 3 \, b d^{2} x e^{\left (-b x - a\right )} - 3 \, b d^{2} x e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b c d e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b c d e^{\left (b x + a\right )} + 3 \, b c d e^{\left (-b x - a\right )} - 3 \, b c d e^{\left (-3 \, b x - 3 \, a\right )} - d^{2} e^{\left (3 \, b x + 3 \, a\right )} + 3 \, d^{2} e^{\left (b x + a\right )} - 3 \, d^{2} e^{\left (-b x - a\right )} + d^{2} e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

1/16*(9*b^2*d^2*x^2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 3*b^2*d^2*x^ 
2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*b^2*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e 
^(-a + b*c/d) - 9*b^2*d^2*x^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 
18*b^2*c*d*x*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 6*b^2*c*d*x*Ei((b*d 
*x + b*c)/d)*e^(a - b*c/d) + 6*b^2*c*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/ 
d) - 18*b^2*c*d*x*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 9*b^2*c^2*Ei 
(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 3*b^2*c^2*Ei((b*d*x + b*c)/d)*e^(a 
 - b*c/d) + 3*b^2*c^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 9*b^2*c^2*Ei(- 
3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) - 3*b*d^2*x*e^(3*b*x + 3*a) + 3*b*d^ 
2*x*e^(b*x + a) + 3*b*d^2*x*e^(-b*x - a) - 3*b*d^2*x*e^(-3*b*x - 3*a) - 3* 
b*c*d*e^(3*b*x + 3*a) + 3*b*c*d*e^(b*x + a) + 3*b*c*d*e^(-b*x - a) - 3*b*c 
*d*e^(-3*b*x - 3*a) - d^2*e^(3*b*x + 3*a) + 3*d^2*e^(b*x + a) - 3*d^2*e^(- 
b*x - a) + d^2*e^(-3*b*x - 3*a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {e^{4 a} \left (\int \frac {e^{3 b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right )-3 e^{2 a} \left (\int \frac {e^{b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right )-e^{a} \left (\int \frac {1}{e^{3 b x +3 a} c^{3}+3 e^{3 b x +3 a} c^{2} d x +3 e^{3 b x +3 a} c \,d^{2} x^{2}+e^{3 b x +3 a} d^{3} x^{3}}d x \right )+3 \left (\int \frac {1}{e^{b x} c^{3}+3 e^{b x} c^{2} d x +3 e^{b x} c \,d^{2} x^{2}+e^{b x} d^{3} x^{3}}d x \right )}{8 e^{a}} \] Input:

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

Output:

(e**(4*a)*int(e**(3*b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x 
) - 3*e**(2*a)*int(e**(b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 
),x) - e**a*int(1/(e**(3*a + 3*b*x)*c**3 + 3*e**(3*a + 3*b*x)*c**2*d*x + 3 
*e**(3*a + 3*b*x)*c*d**2*x**2 + e**(3*a + 3*b*x)*d**3*x**3),x) + 3*int(1/( 
e**(b*x)*c**3 + 3*e**(b*x)*c**2*d*x + 3*e**(b*x)*c*d**2*x**2 + e**(b*x)*d* 
*3*x**3),x))/(8*e**a)