\(\int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx\) [96]

Optimal result

Integrand size = 21, antiderivative size = 98 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 i a d^3 \sinh (e+f x)}{f^4}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3398, 3377, 2717} \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}+\frac {a (c+d x)^4}{4 d}-\frac {6 i a d^3 \sinh (e+f x)}{f^4} \]

[In]

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

[Out]

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

Rule 2717

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

Rule 3377

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

Rule 3398

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^3+i a (c+d x)^3 \sinh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+(i a) \int (c+d x)^3 \sinh (e+f x) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {(3 i a d) \int (c+d x)^2 \cosh (e+f x) \, dx}{f} \\ & = \frac {a (c+d x)^4}{4 d}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {\left (6 i a d^2\right ) \int (c+d x) \sinh (e+f x) \, dx}{f^2} \\ & = \frac {a (c+d x)^4}{4 d}+\frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}-\frac {\left (6 i a d^3\right ) \int \cosh (e+f x) \, dx}{f^3} \\ & = \frac {a (c+d x)^4}{4 d}+\frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 i a d^3 \sinh (e+f x)}{f^4}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {a \left (f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+4 i f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \cosh (e+f x)-12 i d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \sinh (e+f x)\right )}{4 f^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\(\frac {\left (i f \left (d x +c \right ) \left (\left (d x +c \right )^{2} f^{2}+6 d^{2}\right ) \cosh \left (f x +e \right )-3 i d \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \sinh \left (f x +e \right )+f \left (x \left (\frac {1}{2} d^{2} x^{2}+c d x +c^{2}\right ) \left (\frac {d x}{2}+c \right ) f^{3}+i c^{3} f^{2}+6 i c \,d^{2}\right )\right ) a}{f^{4}}\)	\(112\)
risch	\(\frac {a \,d^{3} x^{4}}{4}+a \,d^{2} c \,x^{3}+\frac {3 a d \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {a \,c^{4}}{4 d}+\frac {i a \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{2 f^{4}}+\frac {i a \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{2 f^{4}}\)	\(252\)
parts	\(\frac {a \left (d x +c \right )^{4}}{4 d}+\frac {i a \left (\frac {d^{3} \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{3} e \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d^{2} c \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {6 d^{2} e c \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} \cosh \left (f x +e \right )}{f}+c^{3} \cosh \left (f x +e \right )\right )}{f}\)	\(325\)
derivativedivides	\(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+i c^{3} a \cosh \left (f x +e \right )-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}+\frac {3 i d^{3} e^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 i d^{2} e^{2} c a \cosh \left (f x +e \right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 i d^{2} c a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {i d^{3} e^{3} a \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {i d^{3} a \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}+\frac {3 i d \,c^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}-\frac {3 i d^{3} e a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 i d e \,c^{2} a \cosh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )-\frac {6 i d^{2} e c a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}}{f}\)	\(494\)
default	\(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+i c^{3} a \cosh \left (f x +e \right )-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}+\frac {3 i d^{3} e^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 i d^{2} e^{2} c a \cosh \left (f x +e \right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 i d^{2} c a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {i d^{3} e^{3} a \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {i d^{3} a \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}+\frac {3 i d \,c^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}-\frac {3 i d^{3} e a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 i d e \,c^{2} a \cosh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )-\frac {6 i d^{2} e c a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}}{f}\)	\(494\)

[In]

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

[Out]

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

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. 284 vs. \(2 (88) = 176\).

Time = 0.25 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.90 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {{\left (2 i \, a d^{3} f^{3} x^{3} + 2 i \, a c^{3} f^{3} + 6 i \, a c^{2} d f^{2} + 12 i \, a c d^{2} f + 12 i \, a d^{3} - 6 \, {\left (-i \, a c d^{2} f^{3} - i \, a d^{3} f^{2}\right )} x^{2} - 6 \, {\left (-i \, a c^{2} d f^{3} - 2 i \, a c d^{2} f^{2} - 2 i \, a d^{3} f\right )} x - 2 \, {\left (-i \, a d^{3} f^{3} x^{3} - i \, a c^{3} f^{3} + 3 i \, a c^{2} d f^{2} - 6 i \, a c d^{2} f + 6 i \, a d^{3} + 3 \, {\left (-i \, a c d^{2} f^{3} + i \, a d^{3} f^{2}\right )} x^{2} + 3 \, {\left (-i \, a c^{2} d f^{3} + 2 i \, a c d^{2} f^{2} - 2 i \, a d^{3} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{4 \, f^{4}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 517, normalized size of antiderivative = 5.28 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=a c^{3} x + \frac {3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac {a d^{3} x^{4}}{4} + \begin {cases} \frac {\left (\left (2 i a c^{3} f^{7} + 6 i a c^{2} d f^{7} x + 6 i a c^{2} d f^{6} + 6 i a c d^{2} f^{7} x^{2} + 12 i a c d^{2} f^{6} x + 12 i a c d^{2} f^{5} + 2 i a d^{3} f^{7} x^{3} + 6 i a d^{3} f^{6} x^{2} + 12 i a d^{3} f^{5} x + 12 i a d^{3} f^{4}\right ) e^{- f x} + \left (2 i a c^{3} f^{7} e^{2 e} + 6 i a c^{2} d f^{7} x e^{2 e} - 6 i a c^{2} d f^{6} e^{2 e} + 6 i a c d^{2} f^{7} x^{2} e^{2 e} - 12 i a c d^{2} f^{6} x e^{2 e} + 12 i a c d^{2} f^{5} e^{2 e} + 2 i a d^{3} f^{7} x^{3} e^{2 e} - 6 i a d^{3} f^{6} x^{2} e^{2 e} + 12 i a d^{3} f^{5} x e^{2 e} - 12 i a d^{3} f^{4} e^{2 e}\right ) e^{f x}\right ) e^{- e}}{4 f^{8}} & \text {for}\: f^{8} e^{e} \neq 0 \\\frac {x^{4} \left (i a d^{3} e^{2 e} - i a d^{3}\right ) e^{- e}}{8} + \frac {x^{3} \left (i a c d^{2} e^{2 e} - i a c d^{2}\right ) e^{- e}}{2} + \frac {x^{2} \cdot \left (3 i a c^{2} d e^{2 e} - 3 i a c^{2} d\right ) e^{- e}}{4} + \frac {x \left (i a c^{3} e^{2 e} - i a c^{3}\right ) e^{- e}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

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. 235 vs. \(2 (88) = 176\).

Time = 0.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.40 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {3}{2} i \, a c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {3}{2} i \, a c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {1}{2} i \, a d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {i \, a c^{3} \cosh \left (f x + e\right )}{f} \]

[In]

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

[Out]

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

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. 262 vs. \(2 (88) = 176\).

Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.67 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x - \frac {{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x + 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} + 6 i \, a c d^{2} f^{2} x + 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f + 6 i \, a d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac {{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x - 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} - 6 i \, a c d^{2} f^{2} x - 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f - 6 i \, a d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.00 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {\mathrm {cosh}\left (e+f\,x\right )\,\left (a\,c^3\,f^2+6\,a\,c\,d^2\right )\,1{}\mathrm {i}}{f^3}-\frac {\mathrm {sinh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )\,3{}\mathrm {i}}{f^4}+\frac {a\,d^3\,x^4}{4}+a\,c^3\,x+\frac {x\,\mathrm {cosh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )\,3{}\mathrm {i}}{f^3}+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3+\frac {a\,d^3\,x^3\,\mathrm {cosh}\left (e+f\,x\right )\,1{}\mathrm {i}}{f}-\frac {a\,d^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )\,3{}\mathrm {i}}{f^2}-\frac {a\,c\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )\,6{}\mathrm {i}}{f^2}+\frac {a\,c\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )\,3{}\mathrm {i}}{f} \]

[In]

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

[Out]

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