\(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [266]

Optimal result

Integrand size = 31, antiderivative size = 157 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i (e+f x)^2}{4 a d}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac {2 f^2 \sinh (c+d x)}{a d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}+\frac {i f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d^2}-\frac {i f^2 \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \sinh ^2(c+d x)}{2 a d} \] Output:

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

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.63 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-32 d f (e+f x) \cosh (c+d x)-2 i \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))+8 \left (2 \left (2 f^2+d^2 (e+f x)^2\right )+i d f (e+f x) \cosh (c+d x)\right ) \sinh (c+d x)}{16 a d^3} \] Input:

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

Output:

(-32*d*f*(e + f*x)*Cosh[c + d*x] - (2*I)*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[2* 
(c + d*x)] + 8*(2*(2*f^2 + d^2*(e + f*x)^2) + I*d*f*(e + f*x)*Cosh[c + d*x 
])*Sinh[c + d*x])/(16*a*d^3)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {6097, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 5969, 3042, 25, 3791, 17}

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.

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

\(\Big \downarrow \) 6097

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

Input:

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

Output:

(((e + f*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I*(e + f*x)*Cosh[c + d*x])/d - 
 (I*f*Sinh[c + d*x])/d^2))/d)/a - (I*(((e + f*x)^2*Sinh[c + d*x]^2)/(2*d) 
+ (f*((e + f*x)^2/(4*f) - ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + 
(f*Sinh[c + d*x]^2)/(4*d^2)))/d))/a
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

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 3042

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

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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

rule 5969

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

rule 6097

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

Maple [A] (verified)

Time = 14.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.54

method	result	size
risch	\(-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 d \,f^{2} x -2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 d^{3} a}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}-2 d \,f^{2} x -2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{3} a}-\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}+2 d \,f^{2} x +2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{3} a}-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 d \,f^{2} x +2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{3}}\)	\(241\)

Input:

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

Output:

-1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d^2*e^2-2*d*f^2*x-2*d*e*f+f^2)/d^3/a* 
exp(2*d*x+2*c)+1/2*(d^2*f^2*x^2+2*d^2*e*f*x+d^2*e^2-2*d*f^2*x-2*d*e*f+2*f^ 
2)/d^3/a*exp(d*x+c)-1/2*(d^2*f^2*x^2+2*d^2*e*f*x+d^2*e^2+2*d*f^2*x+2*d*e*f 
+2*f^2)/d^3/a*exp(-d*x-c)-1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d^2*e^2+2*d* 
f^2*x+2*d*e*f+f^2)/a/d^3*exp(-2*d*x-2*c)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.45 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} - 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f + i \, d f^{2}\right )} x + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f - i \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} - 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f - d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} + 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f + d f^{2}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \] Input:

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

Output:

1/16*(-2*I*d^2*f^2*x^2 - 2*I*d^2*e^2 - 2*I*d*e*f - I*f^2 - 2*(2*I*d^2*e*f 
+ I*d*f^2)*x + (-2*I*d^2*f^2*x^2 - 2*I*d^2*e^2 + 2*I*d*e*f - I*f^2 - 2*(2* 
I*d^2*e*f - I*d*f^2)*x)*e^(4*d*x + 4*c) + 8*(d^2*f^2*x^2 + d^2*e^2 - 2*d*e 
*f + 2*f^2 + 2*(d^2*e*f - d*f^2)*x)*e^(3*d*x + 3*c) - 8*(d^2*f^2*x^2 + d^2 
*e^2 + 2*d*e*f + 2*f^2 + 2*(d^2*e*f + d*f^2)*x)*e^(d*x + c))*e^(-2*d*x - 2 
*c)/(a*d^3)
 

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (138) = 276\).

Time = 0.41 (sec) , antiderivative size = 631, normalized size of antiderivative = 4.02 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (\left (- 512 a^{3} d^{11} e^{2} e^{2 c} - 1024 a^{3} d^{11} e f x e^{2 c} - 512 a^{3} d^{11} f^{2} x^{2} e^{2 c} - 1024 a^{3} d^{10} e f e^{2 c} - 1024 a^{3} d^{10} f^{2} x e^{2 c} - 1024 a^{3} d^{9} f^{2} e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{11} e^{2} e^{4 c} + 1024 a^{3} d^{11} e f x e^{4 c} + 512 a^{3} d^{11} f^{2} x^{2} e^{4 c} - 1024 a^{3} d^{10} e f e^{4 c} - 1024 a^{3} d^{10} f^{2} x e^{4 c} + 1024 a^{3} d^{9} f^{2} e^{4 c}\right ) e^{d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{c} - 256 i a^{3} d^{11} e f x e^{c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{c} - 128 i a^{3} d^{10} e f e^{c} - 128 i a^{3} d^{10} f^{2} x e^{c} - 64 i a^{3} d^{9} f^{2} e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{5 c} - 256 i a^{3} d^{11} e f x e^{5 c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{5 c} + 128 i a^{3} d^{10} e f e^{5 c} + 128 i a^{3} d^{10} f^{2} x e^{5 c} - 64 i a^{3} d^{9} f^{2} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{12}} & \text {for}\: a^{4} d^{12} e^{3 c} \neq 0 \\\frac {x^{3} \left (- i f^{2} e^{4 c} + 2 f^{2} e^{3 c} + 2 f^{2} e^{c} + i f^{2}\right ) e^{- 2 c}}{12 a} + \frac {x^{2} \left (- i e f e^{4 c} + 2 e f e^{3 c} + 2 e f e^{c} + i e f\right ) e^{- 2 c}}{4 a} + \frac {x \left (- i e^{2} e^{4 c} + 2 e^{2} e^{3 c} + 2 e^{2} e^{c} + i e^{2}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((((-512*a**3*d**11*e**2*exp(2*c) - 1024*a**3*d**11*e*f*x*exp(2*c 
) - 512*a**3*d**11*f**2*x**2*exp(2*c) - 1024*a**3*d**10*e*f*exp(2*c) - 102 
4*a**3*d**10*f**2*x*exp(2*c) - 1024*a**3*d**9*f**2*exp(2*c))*exp(-d*x) + ( 
512*a**3*d**11*e**2*exp(4*c) + 1024*a**3*d**11*e*f*x*exp(4*c) + 512*a**3*d 
**11*f**2*x**2*exp(4*c) - 1024*a**3*d**10*e*f*exp(4*c) - 1024*a**3*d**10*f 
**2*x*exp(4*c) + 1024*a**3*d**9*f**2*exp(4*c))*exp(d*x) + (-128*I*a**3*d** 
11*e**2*exp(c) - 256*I*a**3*d**11*e*f*x*exp(c) - 128*I*a**3*d**11*f**2*x** 
2*exp(c) - 128*I*a**3*d**10*e*f*exp(c) - 128*I*a**3*d**10*f**2*x*exp(c) - 
64*I*a**3*d**9*f**2*exp(c))*exp(-2*d*x) + (-128*I*a**3*d**11*e**2*exp(5*c) 
 - 256*I*a**3*d**11*e*f*x*exp(5*c) - 128*I*a**3*d**11*f**2*x**2*exp(5*c) + 
 128*I*a**3*d**10*e*f*exp(5*c) + 128*I*a**3*d**10*f**2*x*exp(5*c) - 64*I*a 
**3*d**9*f**2*exp(5*c))*exp(2*d*x))*exp(-3*c)/(1024*a**4*d**12), Ne(a**4*d 
**12*exp(3*c), 0)), (x**3*(-I*f**2*exp(4*c) + 2*f**2*exp(3*c) + 2*f**2*exp 
(c) + I*f**2)*exp(-2*c)/(12*a) + x**2*(-I*e*f*exp(4*c) + 2*e*f*exp(3*c) + 
2*e*f*exp(c) + I*e*f)*exp(-2*c)/(4*a) + x*(-I*e**2*exp(4*c) + 2*e**2*exp(3 
*c) + 2*e**2*exp(c) + I*e**2)*exp(-2*c)/(4*a), True))
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

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. 338 vs. \(2 (141) = 282\).

Time = 0.13 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {{\left (2 i \, d^{2} f^{2} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d^{2} f^{2} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d^{2} f^{2} x^{2} e^{\left (d x + c\right )} + 2 i \, d^{2} f^{2} x^{2} + 4 i \, d^{2} e f x e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{2} e f x e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{2} e f x e^{\left (d x + c\right )} + 4 i \, d^{2} e f x + 2 i \, d^{2} e^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, d f^{2} x e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d^{2} e^{2} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d f^{2} x e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d^{2} e^{2} e^{\left (d x + c\right )} + 16 \, d f^{2} x e^{\left (d x + c\right )} + 2 i \, d^{2} e^{2} + 2 i \, d f^{2} x - 2 i \, d e f e^{\left (4 \, d x + 4 \, c\right )} + 16 \, d e f e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d e f e^{\left (d x + c\right )} + 2 i \, d e f + i \, f^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, f^{2} e^{\left (d x + c\right )} + i \, f^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \] Input:

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

Output:

-1/16*(2*I*d^2*f^2*x^2*e^(4*d*x + 4*c) - 8*d^2*f^2*x^2*e^(3*d*x + 3*c) + 8 
*d^2*f^2*x^2*e^(d*x + c) + 2*I*d^2*f^2*x^2 + 4*I*d^2*e*f*x*e^(4*d*x + 4*c) 
 - 16*d^2*e*f*x*e^(3*d*x + 3*c) + 16*d^2*e*f*x*e^(d*x + c) + 4*I*d^2*e*f*x 
 + 2*I*d^2*e^2*e^(4*d*x + 4*c) - 2*I*d*f^2*x*e^(4*d*x + 4*c) - 8*d^2*e^2*e 
^(3*d*x + 3*c) + 16*d*f^2*x*e^(3*d*x + 3*c) + 8*d^2*e^2*e^(d*x + c) + 16*d 
*f^2*x*e^(d*x + c) + 2*I*d^2*e^2 + 2*I*d*f^2*x - 2*I*d*e*f*e^(4*d*x + 4*c) 
 + 16*d*e*f*e^(3*d*x + 3*c) + 16*d*e*f*e^(d*x + c) + 2*I*d*e*f + I*f^2*e^( 
4*d*x + 4*c) - 16*f^2*e^(3*d*x + 3*c) + 16*f^2*e^(d*x + c) + I*f^2)*e^(-2* 
d*x - 2*c)/(a*d^3)
 

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.73 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx={\mathrm {e}}^{c+d\,x}\,\left (\frac {d^2\,e^2-2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}-\frac {f\,x\,\left (f-d\,e\right )}{a\,d^2}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^2\,e^2+2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}+\frac {f\,x\,\left (f+d\,e\right )}{a\,d^2}\right ) \] Input:

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

Output:

exp(c + d*x)*((2*f^2 + d^2*e^2 - 2*d*e*f)/(2*a*d^3) + (f^2*x^2)/(2*a*d) - 
(f*x*(f - d*e))/(a*d^2)) - exp(- 2*c - 2*d*x)*(((f^2 + 2*d^2*e^2 + 2*d*e*f 
)*1i)/(16*a*d^3) + (f^2*x^2*1i)/(8*a*d) + (f*x*(f + 2*d*e)*1i)/(8*a*d^2)) 
- exp(2*c + 2*d*x)*(((f^2 + 2*d^2*e^2 - 2*d*e*f)*1i)/(16*a*d^3) + (f^2*x^2 
*1i)/(8*a*d) - (f*x*(f - 2*d*e)*1i)/(8*a*d^2)) - exp(- c - d*x)*((2*f^2 + 
d^2*e^2 + 2*d*e*f)/(2*a*d^3) + (f^2*x^2)/(2*a*d) + (f*x*(f + d*e))/(a*d^2) 
)
 

Reduce [F]

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

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

Output:

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