\(\int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [259]

Optimal result

Integrand size = 31, antiderivative size = 108 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2} \] Output:

1/4*(f*x+e)^4/a/f-6*I*f^2*(f*x+e)*cosh(d*x+c)/a/d^3-I*(f*x+e)^3*cosh(d*x+c 
)/a/d+6*I*f^3*sinh(d*x+c)/a/d^4+3*I*f*(f*x+e)^2*sinh(d*x+c)/a/d^2
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 i d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right ) \cosh (c+d x)+12 i f \left (2 f^2+d^2 (e+f x)^2\right ) \sinh (c+d x)}{4 a d^4} \] Input:

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

Output:

(d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - (4*I)*d*(e + f*x)*(6* 
f^2 + d^2*(e + f*x)^2)*Cosh[c + d*x] + (12*I)*f*(2*f^2 + d^2*(e + f*x)^2)* 
Sinh[c + d*x])/(4*a*d^4)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {6097, 17, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117}

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.

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

\(\Big \downarrow \) 6097

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

Input:

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

Output:

(e + f*x)^4/(4*a*f) - ((I*(e + f*x)^3*Cosh[c + d*x])/d - ((3*I)*f*(((e + f 
*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I*(e + f*x)*Cosh[c + d*x])/d - (I*f*Si 
nh[c + d*x])/d^2))/d))/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 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 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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (102 ) = 204\).

Time = 13.60 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.46

method	result	size
risch	\(\frac {f^{3} x^{4}}{4 a}+\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {e^{4}}{4 a f}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}\)	\(266\)
derivativedivides	\(-\frac {-6 i c d e \,f^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )-3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )+i d^{3} e^{3} \cosh \left (d x +c \right )+3 i d e \,f^{2} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )+3 i d^{2} e^{2} f \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )+3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )-i c^{3} f^{3} \cosh \left (d x +c \right )+i f^{3} \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )-3 i c \,f^{3} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )+3 i c^{2} f^{3} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )+c^{3} f^{3} \left (d x +c \right )-3 c^{2} d e \,f^{2} \left (d x +c \right )-\frac {3 c^{2} f^{3} \left (d x +c \right )^{2}}{2}+3 c \,d^{2} e^{2} f \left (d x +c \right )+3 c d e \,f^{2} \left (d x +c \right )^{2}+c \,f^{3} \left (d x +c \right )^{3}-d^{3} e^{3} \left (d x +c \right )-\frac {3 d^{2} e^{2} f \left (d x +c \right )^{2}}{2}-d e \,f^{2} \left (d x +c \right )^{3}-\frac {f^{3} \left (d x +c \right )^{4}}{4}}{d^{4} a}\)	\(448\)
default	\(-\frac {-6 i c d e \,f^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )-3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )+i d^{3} e^{3} \cosh \left (d x +c \right )+3 i d e \,f^{2} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )+3 i d^{2} e^{2} f \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )+3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )-i c^{3} f^{3} \cosh \left (d x +c \right )+i f^{3} \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )-3 i c \,f^{3} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )+3 i c^{2} f^{3} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )+c^{3} f^{3} \left (d x +c \right )-3 c^{2} d e \,f^{2} \left (d x +c \right )-\frac {3 c^{2} f^{3} \left (d x +c \right )^{2}}{2}+3 c \,d^{2} e^{2} f \left (d x +c \right )+3 c d e \,f^{2} \left (d x +c \right )^{2}+c \,f^{3} \left (d x +c \right )^{3}-d^{3} e^{3} \left (d x +c \right )-\frac {3 d^{2} e^{2} f \left (d x +c \right )^{2}}{2}-d e \,f^{2} \left (d x +c \right )^{3}-\frac {f^{3} \left (d x +c \right )^{4}}{4}}{d^{4} a}\)	\(448\)

Input:

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

Output:

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

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. 263 vs. \(2 (98) = 196\).

Time = 0.09 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.44 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-2 i \, d^{3} f^{3} x^{3} - 2 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 12 i \, d e f^{2} - 12 i \, f^{3} - 6 \, {\left (i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} - 6 \, {\left (i \, d^{3} e^{2} f + 2 i \, d^{2} e f^{2} + 2 i \, d f^{3}\right )} x - 2 \, {\left (i \, d^{3} f^{3} x^{3} + i \, d^{3} e^{3} - 3 i \, d^{2} e^{2} f + 6 i \, d e f^{2} - 6 i \, f^{3} + 3 \, {\left (i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} + 3 \, {\left (i \, d^{3} e^{2} f - 2 i \, d^{2} e f^{2} + 2 i \, d f^{3}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (d^{4} f^{3} x^{4} + 4 \, d^{4} e f^{2} x^{3} + 6 \, d^{4} e^{2} f x^{2} + 4 \, d^{4} e^{3} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-d x - c\right )}}{4 \, a d^{4}} \] Input:

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

Output:

1/4*(-2*I*d^3*f^3*x^3 - 2*I*d^3*e^3 - 6*I*d^2*e^2*f - 12*I*d*e*f^2 - 12*I* 
f^3 - 6*(I*d^3*e*f^2 + I*d^2*f^3)*x^2 - 6*(I*d^3*e^2*f + 2*I*d^2*e*f^2 + 2 
*I*d*f^3)*x - 2*(I*d^3*f^3*x^3 + I*d^3*e^3 - 3*I*d^2*e^2*f + 6*I*d*e*f^2 - 
 6*I*f^3 + 3*(I*d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(I*d^3*e^2*f - 2*I*d^2*e*f^ 
2 + 2*I*d*f^3)*x)*e^(2*d*x + 2*c) + (d^4*f^3*x^4 + 4*d^4*e*f^2*x^3 + 6*d^4 
*e^2*f*x^2 + 4*d^4*e^3*x)*e^(d*x + c))*e^(-d*x - c)/(a*d^4)
 

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 518, normalized size of antiderivative = 4.80 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (\left (- 2 i a d^{7} e^{3} - 6 i a d^{7} e^{2} f x - 6 i a d^{7} e f^{2} x^{2} - 2 i a d^{7} f^{3} x^{3} - 6 i a d^{6} e^{2} f - 12 i a d^{6} e f^{2} x - 6 i a d^{6} f^{3} x^{2} - 12 i a d^{5} e f^{2} - 12 i a d^{5} f^{3} x - 12 i a d^{4} f^{3}\right ) e^{- d x} + \left (- 2 i a d^{7} e^{3} e^{2 c} - 6 i a d^{7} e^{2} f x e^{2 c} - 6 i a d^{7} e f^{2} x^{2} e^{2 c} - 2 i a d^{7} f^{3} x^{3} e^{2 c} + 6 i a d^{6} e^{2} f e^{2 c} + 12 i a d^{6} e f^{2} x e^{2 c} + 6 i a d^{6} f^{3} x^{2} e^{2 c} - 12 i a d^{5} e f^{2} e^{2 c} - 12 i a d^{5} f^{3} x e^{2 c} + 12 i a d^{4} f^{3} e^{2 c}\right ) e^{d x}\right ) e^{- c}}{4 a^{2} d^{8}} & \text {for}\: a^{2} d^{8} e^{c} \neq 0 \\\frac {x^{4} \left (- i f^{3} e^{2 c} + i f^{3}\right ) e^{- c}}{8 a} + \frac {x^{3} \left (- i e f^{2} e^{2 c} + i e f^{2}\right ) e^{- c}}{2 a} + \frac {x^{2} \left (- 3 i e^{2} f e^{2 c} + 3 i e^{2} f\right ) e^{- c}}{4 a} + \frac {x \left (- i e^{3} e^{2 c} + i e^{3}\right ) e^{- c}}{2 a} & \text {otherwise} \end {cases} + \frac {e^{3} x}{a} + \frac {3 e^{2} f x^{2}}{2 a} + \frac {e f^{2} x^{3}}{a} + \frac {f^{3} x^{4}}{4 a} \] Input:

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

Output:

Piecewise((((-2*I*a*d**7*e**3 - 6*I*a*d**7*e**2*f*x - 6*I*a*d**7*e*f**2*x* 
*2 - 2*I*a*d**7*f**3*x**3 - 6*I*a*d**6*e**2*f - 12*I*a*d**6*e*f**2*x - 6*I 
*a*d**6*f**3*x**2 - 12*I*a*d**5*e*f**2 - 12*I*a*d**5*f**3*x - 12*I*a*d**4* 
f**3)*exp(-d*x) + (-2*I*a*d**7*e**3*exp(2*c) - 6*I*a*d**7*e**2*f*x*exp(2*c 
) - 6*I*a*d**7*e*f**2*x**2*exp(2*c) - 2*I*a*d**7*f**3*x**3*exp(2*c) + 6*I* 
a*d**6*e**2*f*exp(2*c) + 12*I*a*d**6*e*f**2*x*exp(2*c) + 6*I*a*d**6*f**3*x 
**2*exp(2*c) - 12*I*a*d**5*e*f**2*exp(2*c) - 12*I*a*d**5*f**3*x*exp(2*c) + 
 12*I*a*d**4*f**3*exp(2*c))*exp(d*x))*exp(-c)/(4*a**2*d**8), Ne(a**2*d**8* 
exp(c), 0)), (x**4*(-I*f**3*exp(2*c) + I*f**3)*exp(-c)/(8*a) + x**3*(-I*e* 
f**2*exp(2*c) + I*e*f**2)*exp(-c)/(2*a) + x**2*(-3*I*e**2*f*exp(2*c) + 3*I 
*e**2*f)*exp(-c)/(4*a) + x*(-I*e**3*exp(2*c) + I*e**3)*exp(-c)/(2*a), True 
)) + e**3*x/a + 3*e**2*f*x**2/(2*a) + e*f**2*x**3/a + f**3*x**4/(4*a)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (98) = 196\).

Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.45 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3}{2} \, e^{2} f {\left (\frac {2 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} - \frac {i \, d^{2} x^{2} e^{c} + i \, d x e^{c} - {\left (-i \, d x e^{\left (3 \, c\right )} + i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (d x + 1\right )} e^{\left (-d x\right )} + i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}}\right )} + \frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )}}{a d} - \frac {i \, e^{\left (d x + c\right )}}{a d} - \frac {i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac {{\left (2 \, d^{3} x^{3} e^{c} + 3 \, {\left (-i \, d^{2} x^{2} e^{\left (2 \, c\right )} + 2 i \, d x e^{\left (2 \, c\right )} - 2 i \, e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 3 \, {\left (-i \, d^{2} x^{2} - 2 i \, d x - 2 i\right )} e^{\left (-d x\right )}\right )} e f^{2} e^{\left (-c\right )}}{2 \, a d^{3}} + \frac {{\left (d^{4} x^{4} e^{c} + 2 \, {\left (-i \, d^{3} x^{3} e^{\left (2 \, c\right )} + 3 i \, d^{2} x^{2} e^{\left (2 \, c\right )} - 6 i \, d x e^{\left (2 \, c\right )} + 6 i \, e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 2 \, {\left (-i \, d^{3} x^{3} - 3 i \, d^{2} x^{2} - 6 i \, d x - 6 i\right )} e^{\left (-d x\right )}\right )} f^{3} e^{\left (-c\right )}}{4 \, a d^{4}} \] Input:

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

Output:

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

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. 355 vs. \(2 (98) = 196\).

Time = 0.12 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.29 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (d^{4} f^{3} x^{4} e^{\left (d x + c\right )} + 4 \, d^{4} e f^{2} x^{3} e^{\left (d x + c\right )} - 2 i \, d^{3} f^{3} x^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, d^{4} e^{2} f x^{2} e^{\left (d x + c\right )} - 2 i \, d^{3} f^{3} x^{3} - 6 i \, d^{3} e f^{2} x^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, d^{4} e^{3} x e^{\left (d x + c\right )} - 6 i \, d^{3} e f^{2} x^{2} - 6 i \, d^{3} e^{2} f x e^{\left (2 \, d x + 2 \, c\right )} + 6 i \, d^{2} f^{3} x^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d^{3} e^{2} f x - 6 i \, d^{2} f^{3} x^{2} - 2 i \, d^{3} e^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 i \, d^{2} e f^{2} x e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, d^{3} e^{3} - 12 i \, d^{2} e f^{2} x + 6 i \, d^{2} e^{2} f e^{\left (2 \, d x + 2 \, c\right )} - 12 i \, d f^{3} x e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d^{2} e^{2} f - 12 i \, d f^{3} x - 12 i \, d e f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 12 i \, d e f^{2} + 12 i \, f^{3} e^{\left (2 \, d x + 2 \, c\right )} - 12 i \, f^{3}\right )} e^{\left (-d x - c\right )}}{4 \, a d^{4}} \] Input:

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

Output:

1/4*(d^4*f^3*x^4*e^(d*x + c) + 4*d^4*e*f^2*x^3*e^(d*x + c) - 2*I*d^3*f^3*x 
^3*e^(2*d*x + 2*c) + 6*d^4*e^2*f*x^2*e^(d*x + c) - 2*I*d^3*f^3*x^3 - 6*I*d 
^3*e*f^2*x^2*e^(2*d*x + 2*c) + 4*d^4*e^3*x*e^(d*x + c) - 6*I*d^3*e*f^2*x^2 
 - 6*I*d^3*e^2*f*x*e^(2*d*x + 2*c) + 6*I*d^2*f^3*x^2*e^(2*d*x + 2*c) - 6*I 
*d^3*e^2*f*x - 6*I*d^2*f^3*x^2 - 2*I*d^3*e^3*e^(2*d*x + 2*c) + 12*I*d^2*e* 
f^2*x*e^(2*d*x + 2*c) - 2*I*d^3*e^3 - 12*I*d^2*e*f^2*x + 6*I*d^2*e^2*f*e^( 
2*d*x + 2*c) - 12*I*d*f^3*x*e^(2*d*x + 2*c) - 6*I*d^2*e^2*f - 12*I*d*f^3*x 
 - 12*I*d*e*f^2*e^(2*d*x + 2*c) - 12*I*d*e*f^2 + 12*I*f^3*e^(2*d*x + 2*c) 
- 12*I*f^3)*e^(-d*x - c)/(a*d^4)
 

Mupad [B] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.49 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx={\mathrm {e}}^{c+d\,x}\,\left (\frac {\left (-d^3\,e^3+3\,d^2\,e^2\,f-6\,d\,e\,f^2+6\,f^3\right )\,1{}\mathrm {i}}{2\,a\,d^4}-\frac {f^3\,x^3\,1{}\mathrm {i}}{2\,a\,d}+\frac {f^2\,x^2\,\left (f-d\,e\right )\,3{}\mathrm {i}}{2\,a\,d^2}-\frac {f\,x\,\left (d^2\,e^2-2\,d\,e\,f+2\,f^2\right )\,3{}\mathrm {i}}{2\,a\,d^3}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {\left (d^3\,e^3+3\,d^2\,e^2\,f+6\,d\,e\,f^2+6\,f^3\right )\,1{}\mathrm {i}}{2\,a\,d^4}+\frac {f^3\,x^3\,1{}\mathrm {i}}{2\,a\,d}+\frac {f^2\,x^2\,\left (f+d\,e\right )\,3{}\mathrm {i}}{2\,a\,d^2}+\frac {f\,x\,\left (d^2\,e^2+2\,d\,e\,f+2\,f^2\right )\,3{}\mathrm {i}}{2\,a\,d^3}\right )+\frac {e^3\,x}{a}+\frac {f^3\,x^4}{4\,a}+\frac {3\,e^2\,f\,x^2}{2\,a}+\frac {e\,f^2\,x^3}{a} \] Input:

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

Output:

exp(c + d*x)*(((6*f^3 - d^3*e^3 + 3*d^2*e^2*f - 6*d*e*f^2)*1i)/(2*a*d^4) - 
 (f^3*x^3*1i)/(2*a*d) + (f^2*x^2*(f - d*e)*3i)/(2*a*d^2) - (f*x*(2*f^2 + d 
^2*e^2 - 2*d*e*f)*3i)/(2*a*d^3)) - exp(- c - d*x)*(((6*f^3 + d^3*e^3 + 3*d 
^2*e^2*f + 6*d*e*f^2)*1i)/(2*a*d^4) + (f^3*x^3*1i)/(2*a*d) + (f^2*x^2*(f + 
 d*e)*3i)/(2*a*d^2) + (f*x*(2*f^2 + d^2*e^2 + 2*d*e*f)*3i)/(2*a*d^3)) + (e 
^3*x)/a + (f^3*x^4)/(4*a) + (3*e^2*f*x^2)/(2*a) + (e*f^2*x^3)/a
 

Reduce [F]

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

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

Output:

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