\(\int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx\) [107]

Optimal result

Integrand size = 23, antiderivative size = 236 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {a^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}+\frac {i a^2 f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}+\frac {i a^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}-\frac {a^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \] Output:

-2*a^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^4/d/(d*x+c)^2-a^2*f^2*cosh(-2*e+2*c*f/ 
d)*Chi(2*c*f/d+2*f*x)/d^3-I*a^2*f^2*Chi(c*f/d+f*x)*sinh(-e+c*f/d)/d^3-4*a^ 
2*f*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^3*sinh(1/2*e+1/4*I*Pi+1/2*f*x)/d^2/(d*x+c 
)+I*a^2*f^2*cosh(-e+c*f/d)*Shi(c*f/d+f*x)/d^3+a^2*f^2*sinh(-2*e+2*c*f/d)*S 
hi(2*c*f/d+2*f*x)/d^3
 

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.84 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {a^2 \left (-4 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+4 i f^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+\frac {d (-3 d-4 i f (c+d x) \cosh (e+f x)+d \cosh (2 (e+f x))-4 i d \sinh (e+f x)+2 c f \sinh (2 (e+f x))+2 d f x \sinh (2 (e+f x)))}{(c+d x)^2}+4 i f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-4 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 d^3} \] Input:

Integrate[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^3,x]
 

Output:

(a^2*(-4*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x))/d] + (4*I) 
*f^2*CoshIntegral[f*(c/d + x)]*Sinh[e - (c*f)/d] + (d*(-3*d - (4*I)*f*(c + 
 d*x)*Cosh[e + f*x] + d*Cosh[2*(e + f*x)] - (4*I)*d*Sinh[e + f*x] + 2*c*f* 
Sinh[2*(e + f*x)] + 2*d*f*x*Sinh[2*(e + f*x)]))/(c + d*x)^2 + (4*I)*f^2*Co 
sh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] - 4*f^2*Sinh[2*e - (2*c*f)/d]*Si 
nhIntegral[(2*f*(c + d*x))/d]))/(4*d^3)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3799, 3042, 3795, 3042, 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.

\(\displaystyle \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+a \sin (i e+i f x))^2}{(c+d x)^3}dx\)

\(\Big \downarrow \) 3799

\(\displaystyle 4 a^2 \int \frac {\sinh ^4\left (\frac {e}{2}+\frac {f x}{2}-\frac {i \pi }{4}\right )}{(c+d x)^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^4}{(c+d x)^3}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle 4 a^2 \left (\frac {2 f^2 \int \frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{c+d x}dx}{d^2}-\frac {3 f^2 \int \frac {\cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{c+d x}dx}{2 d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2}{c+d x}dx}{2 d^2}+\frac {2 f^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^4}{c+d x}dx}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \int \left (\frac {i \sinh (e+f x)}{2 (c+d x)}+\frac {1}{2 (c+d x)}\right )dx}{2 d^2}+\frac {2 f^2 \int \left (-\frac {\cosh (2 e+2 f x)}{8 (c+d x)}+\frac {i \sinh (e+f x)}{2 (c+d x)}+\frac {3}{8 (c+d x)}\right )dx}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \left (\frac {i \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d}+\frac {i \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d}+\frac {\log (c+d x)}{2 d}\right )}{2 d^2}+\frac {2 f^2 \left (\frac {i \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d}-\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{8 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 d}+\frac {i \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d}+\frac {3 \log (c+d x)}{8 d}\right )}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 d (c+d x)^2}\right )\)

Input:

Int[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^3,x]
 

Output:

4*a^2*(-1/2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^4/(d*(c + d*x)^2) - (f*Cosh[e/2 
 + (I/4)*Pi + (f*x)/2]^3*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/(d^2*(c + d*x)) - 
 (3*f^2*(Log[c + d*x]/(2*d) + ((I/2)*CoshIntegral[(c*f)/d + f*x]*Sinh[e - 
(c*f)/d])/d + ((I/2)*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d))/(2 
*d^2) + (2*f^2*(-1/8*(Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x 
])/d + (3*Log[c + d*x])/(8*d) + ((I/2)*CoshIntegral[(c*f)/d + f*x]*Sinh[e 
- (c*f)/d])/d + ((I/2)*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d - 
(Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*d)))/d^2)
 

Defintions of rubi rules used

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 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]
 

rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (214 ) = 428\).

Time = 1.29 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.65

method	result	size
risch	\(-\frac {i a^{2} f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {i a^{2} f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {i a^{2} f^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {3 a^{2}}{4 d \left (d x +c \right )^{2}}-\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}+\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}+\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}+\frac {f^{2} a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}-\frac {i a^{2} f^{3} {\mathrm e}^{-f x -e} x}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {i a^{2} f^{3} {\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a^{2} f^{2} {\mathrm e}^{-f x -e}}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a^{2} f^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}\)	\(625\)

Input:

int((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*I*a^2*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*I*a^2*f^2/d^3*exp(f*x+e)/( 
c*f/d+f*x)-1/2*I*a^2*f^2/d^3*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-3/ 
4*a^2/d/(d*x+c)^2-1/4*f^3*a^2*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c 
^2*f^2)*x-1/4*f^3*a^2*exp(-2*f*x-2*e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2 
)*c+1/8*f^2*a^2*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/2*f^ 
2*a^2/d^3*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)+1/8*f^2*a^2/d^3 
*exp(2*f*x+2*e)/(c*f/d+f*x)^2+1/4*f^2*a^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)+1 
/2*f^2*a^2/d^3*exp(-2*(c*f-d*e)/d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)-1/2*I*a^ 
2*f^3*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x-1/2*I*a^2*f^3*exp( 
-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/2*I*a^2*f^2*exp(-f*x-e)/ 
d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/2*I*a^2*f^2/d^3*exp((c*f-d*e)/d)*Ei( 
1,f*x+e+(c*f-d*e)/d)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (209) = 418\).

Time = 0.11 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.92 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {{\left (2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f - a^{2} d^{2} - {\left (2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (4 \, f x + 4 \, e\right )} + 4 \, {\left (i \, a^{2} d^{2} f x + i \, a^{2} c d f + i \, a^{2} d^{2}\right )} e^{\left (3 \, f x + 3 \, e\right )} + 2 \, {\left (3 \, a^{2} d^{2} + 2 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, {\left (-i \, a^{2} d^{2} f^{2} x^{2} - 2 i \, a^{2} c d f^{2} x - i \, a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + 2 \, {\left (i \, a^{2} d^{2} f^{2} x^{2} + 2 i \, a^{2} c d f^{2} x + i \, a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + 2 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}\right )} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, {\left (i \, a^{2} d^{2} f x + i \, a^{2} c d f - i \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")
 

Output:

-1/8*(2*a^2*d^2*f*x + 2*a^2*c*d*f - a^2*d^2 - (2*a^2*d^2*f*x + 2*a^2*c*d*f 
 + a^2*d^2)*e^(4*f*x + 4*e) + 4*(I*a^2*d^2*f*x + I*a^2*c*d*f + I*a^2*d^2)* 
e^(3*f*x + 3*e) + 2*(3*a^2*d^2 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^ 
2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d)*e^(2*(d*e - c*f)/d) + 2*(-I*a^2*d^2*f^2*x 
^2 - 2*I*a^2*c*d*f^2*x - I*a^2*c^2*f^2)*Ei((d*f*x + c*f)/d)*e^((d*e - c*f) 
/d) + 2*(I*a^2*d^2*f^2*x^2 + 2*I*a^2*c*d*f^2*x + I*a^2*c^2*f^2)*Ei(-(d*f*x 
 + c*f)/d)*e^(-(d*e - c*f)/d) + 2*(a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2 
*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*e^(-2*(d*e - c*f)/d))*e^(2*f*x + 2*e) + 4 
*(I*a^2*d^2*f*x + I*a^2*c*d*f - I*a^2*d^2)*e^(f*x + e))*e^(-2*f*x - 2*e)/( 
d^5*x^2 + 2*c*d^4*x + c^2*d^3)
 

Sympy [F]

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

integrate((a+I*a*sinh(f*x+e))**2/(d*x+c)**3,x)
 

Output:

-a**2*(Integral(sinh(e + f*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3 
*x**3), x) + Integral(-2*I*sinh(e + f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x** 
2 + d**3*x**3), x) + Integral(-1/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3 
*x**3), x))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.87 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + i \, a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")
 

Output:

-1/4*a^2*(1/(d^3*x^2 + 2*c*d^2*x + c^2*d) - e^(-2*e + 2*c*f/d)*exp_integra 
l_e(3, 2*(d*x + c)*f/d)/((d*x + c)^2*d) - e^(2*e - 2*c*f/d)*exp_integral_e 
(3, -2*(d*x + c)*f/d)/((d*x + c)^2*d)) + I*a^2*(e^(-e + c*f/d)*exp_integra 
l_e(3, (d*x + c)*f/d)/((d*x + c)^2*d) - e^(e - c*f/d)*exp_integral_e(3, -( 
d*x + c)*f/d)/((d*x + c)^2*d)) - 1/2*a^2/(d^3*x^2 + 2*c*d^2*x + c^2*d)
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (209) = 418\).

Time = 0.12 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.89 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="giac")
 

Output:

-1/8*(4*a^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) - 4*I*a^2* 
d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*I*a^2*d^2*f^2*x^2*Ei(-(d 
*f*x + c*f)/d)*e^(-e + c*f/d) + 4*a^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e 
^(-2*e + 2*c*f/d) + 8*a^2*c*d*f^2*x*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d 
) - 8*I*a^2*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 8*I*a^2*c*d*f^2* 
x*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 8*a^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f 
)/d)*e^(-2*e + 2*c*f/d) + 4*a^2*c^2*f^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c 
*f/d) - 4*I*a^2*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*I*a^2*c^2*f^ 
2*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 4*a^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/ 
d)*e^(-2*e + 2*c*f/d) - 2*a^2*d^2*f*x*e^(2*f*x + 2*e) + 4*I*a^2*d^2*f*x*e^ 
(f*x + e) + 4*I*a^2*d^2*f*x*e^(-f*x - e) + 2*a^2*d^2*f*x*e^(-2*f*x - 2*e) 
- 2*a^2*c*d*f*e^(2*f*x + 2*e) + 4*I*a^2*c*d*f*e^(f*x + e) + 4*I*a^2*c*d*f* 
e^(-f*x - e) + 2*a^2*c*d*f*e^(-2*f*x - 2*e) - a^2*d^2*e^(2*f*x + 2*e) + 4* 
I*a^2*d^2*e^(f*x + e) - 4*I*a^2*d^2*e^(-f*x - e) - a^2*d^2*e^(-2*f*x - 2*e 
) + 6*a^2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \] Input:

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^3,x)
 

Output:

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^3, x)
                                                                                    
                                                                                    
 

Reduce [F]

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

int((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x)
 

Output:

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