Integrand size = 21, antiderivative size = 98 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 i a d^3 \sinh (e+f x)}{f^4}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2} \] Output:
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
Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {a \left (f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+4 i f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \cosh (e+f x)-12 i d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \sinh (e+f x)\right )}{4 f^4} \] Input:
Integrate[(c + d*x)^3*(a + I*a*Sinh[e + f*x]),x]
Output:
(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)
Time = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3798, 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.
| \(\displaystyle \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 (a+a \sin (i e+i f x))dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a (c+d x)^3+i a (c+d x)^3 \sinh (e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac {3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {i a (c+d x)^3 \cosh (e+f x)}{f}+\frac {a (c+d x)^4}{4 d}-\frac {6 i a d^3 \sinh (e+f x)}{f^4}\) |
Input:
Int[(c + d*x)^3*(a + I*a*Sinh[e + f*x]),x]
Output:
(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
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])
Time = 0.46 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14
| method | result | size |
| parallelrisch | \(\frac {a \left (i \left (d x +c \right ) \left (\left (d x +c \right )^{2} f^{2}+6 d^{2}\right ) f \cosh \left (f x +e \right )-3 i \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) d \sinh \left (f x +e \right )+\left (\left (\frac {d x}{2}+c \right ) x \left (\frac {1}{2} x^{2} d^{2}+c d x +c^{2}\right ) f^{3}+i c^{3} f^{2}+6 i c \,d^{2}\right ) f \right )}{f^{4}}\) | \(112\) |
| 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 | \(\frac {a \left (d x +c \right )^{4}}{4 d}+\frac {i a \left (\frac {d^{3} \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{3} e \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d^{2} c \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {6 d^{2} e c \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} \cosh \left (f x +e \right )}{f}+c^{3} \cosh \left (f x +e \right )\right )}{f}\) | \(325\) |
| orering | \(\frac {\left (d^{5} f^{4} x^{6}+6 c \,d^{4} f^{4} x^{5}+15 c^{2} d^{3} f^{4} x^{4}+20 c^{3} d^{2} f^{4} x^{3}+14 c^{4} d \,f^{4} x^{2}-24 d^{5} f^{2} x^{4}+4 c^{5} f^{4} x -96 c \,d^{4} f^{2} x^{3}-156 c^{2} d^{3} f^{2} x^{2}-120 c^{3} d^{2} f^{2} x -24 c^{4} d \,f^{2}-240 d^{5} x^{2}-480 d^{4} c x -96 d^{3} c^{2}\right ) \left (a +i a \sinh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{2}}+\frac {\left (5 d^{4} f^{2} x^{4}+20 c \,d^{3} f^{2} x^{3}+30 c^{2} d^{2} f^{2} x^{2}+20 c^{3} d \,f^{2} x +2 c^{4} f^{2}+48 d^{4} x^{2}+96 d^{3} c x +12 d^{2} c^{2}\right ) \left (3 \left (d x +c \right )^{2} \left (a +i a \sinh \left (f x +e \right )\right ) d +i \left (d x +c \right )^{3} a f \cosh \left (f x +e \right )\right )}{2 \left (d x +c \right )^{4} f^{4}}-\frac {x \left (f^{2} d^{3} x^{3}+4 c \,d^{2} f^{2} x^{2}+6 c^{2} d \,f^{2} x +4 c^{3} f^{2}+12 d^{3} x +24 d^{2} c \right ) \left (6 \left (d x +c \right ) \left (a +i a \sinh \left (f x +e \right )\right ) d^{2}+6 i \left (d x +c \right )^{2} a f \cosh \left (f x +e \right ) d +i \left (d x +c \right )^{3} a \,f^{2} \sinh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{3}}\) | \(439\) |
| derivativedivides | \(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+i c^{3} a \cosh \left (f x +e \right )-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {i d^{3} e^{3} a \cosh \left (f x +e \right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 i d \,c^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 i d^{3} e^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {3 i d^{3} e a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}-\frac {3 i d e \,c^{2} a \cosh \left (f x +e \right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}+\frac {3 i d^{2} c a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 i d^{2} e^{2} c a \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {6 i d^{2} e c a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+c^{3} a \left (f x +e \right )+\frac {i d^{3} a \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}}{f}\) | \(494\) |
| default | \(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+i c^{3} a \cosh \left (f x +e \right )-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {i d^{3} e^{3} a \cosh \left (f x +e \right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 i d \,c^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 i d^{3} e^{2} a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {3 i d^{3} e a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}-\frac {3 i d e \,c^{2} a \cosh \left (f x +e \right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}+\frac {3 i d^{2} c a \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 i d^{2} e^{2} c a \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {6 i d^{2} e c a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+c^{3} a \left (f x +e \right )+\frac {i d^{3} a \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}}{f}\) | \(494\) |
Input:
int((d*x+c)^3*(a+I*a*sinh(f*x+e)),x,method=_RETURNVERBOSE)
Output:
a*(I*(d*x+c)*((d*x+c)^2*f^2+6*d^2)*f*cosh(f*x+e)-3*I*((d*x+c)^2*f^2+2*d^2) *d*sinh(f*x+e)+((1/2*d*x+c)*x*(1/2*x^2*d^2+c*d*x+c^2)*f^3+I*c^3*f^2+6*I*c* d^2)*f)/f^4
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 = 0.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.90 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {{\left (2 i \, a d^{3} f^{3} x^{3} + 2 i \, a c^{3} f^{3} + 6 i \, a c^{2} d f^{2} + 12 i \, a c d^{2} f + 12 i \, a d^{3} - 6 \, {\left (-i \, a c d^{2} f^{3} - i \, a d^{3} f^{2}\right )} x^{2} - 6 \, {\left (-i \, a c^{2} d f^{3} - 2 i \, a c d^{2} f^{2} - 2 i \, a d^{3} f\right )} x - 2 \, {\left (-i \, a d^{3} f^{3} x^{3} - i \, a c^{3} f^{3} + 3 i \, a c^{2} d f^{2} - 6 i \, a c d^{2} f + 6 i \, a d^{3} + 3 \, {\left (-i \, a c d^{2} f^{3} + i \, a d^{3} f^{2}\right )} x^{2} + 3 \, {\left (-i \, a c^{2} d f^{3} + 2 i \, a c d^{2} f^{2} - 2 i \, a d^{3} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{4 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+I*a*sinh(f*x+e)),x, algorithm="fricas")
Output:
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
Time = 0.34 (sec) , antiderivative size = 517, normalized size of antiderivative = 5.28 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=a c^{3} x + \frac {3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac {a d^{3} x^{4}}{4} + \begin {cases} \frac {\left (\left (2 i a c^{3} f^{7} + 6 i a c^{2} d f^{7} x + 6 i a c^{2} d f^{6} + 6 i a c d^{2} f^{7} x^{2} + 12 i a c d^{2} f^{6} x + 12 i a c d^{2} f^{5} + 2 i a d^{3} f^{7} x^{3} + 6 i a d^{3} f^{6} x^{2} + 12 i a d^{3} f^{5} x + 12 i a d^{3} f^{4}\right ) e^{- f x} + \left (2 i a c^{3} f^{7} e^{2 e} + 6 i a c^{2} d f^{7} x e^{2 e} - 6 i a c^{2} d f^{6} e^{2 e} + 6 i a c d^{2} f^{7} x^{2} e^{2 e} - 12 i a c d^{2} f^{6} x e^{2 e} + 12 i a c d^{2} f^{5} e^{2 e} + 2 i a d^{3} f^{7} x^{3} e^{2 e} - 6 i a d^{3} f^{6} x^{2} e^{2 e} + 12 i a d^{3} f^{5} x e^{2 e} - 12 i a d^{3} f^{4} e^{2 e}\right ) e^{f x}\right ) e^{- e}}{4 f^{8}} & \text {for}\: f^{8} e^{e} \neq 0 \\\frac {x^{4} \left (i a d^{3} e^{2 e} - i a d^{3}\right ) e^{- e}}{8} + \frac {x^{3} \left (i a c d^{2} e^{2 e} - i a c d^{2}\right ) e^{- e}}{2} + \frac {x^{2} \cdot \left (3 i a c^{2} d e^{2 e} - 3 i a c^{2} d\right ) e^{- e}}{4} + \frac {x \left (i a c^{3} e^{2 e} - i a c^{3}\right ) e^{- e}}{2} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*(a+I*a*sinh(f*x+e)),x)
Output:
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))
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 = 0.05 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.40 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {3}{2} i \, a c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {3}{2} i \, a c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {1}{2} i \, a d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {i \, a c^{3} \cosh \left (f x + e\right )}{f} \] Input:
integrate((d*x+c)^3*(a+I*a*sinh(f*x+e)),x, algorithm="maxima")
Output:
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
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 = 0.14 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.67 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x - \frac {{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x + 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} + 6 i \, a c d^{2} f^{2} x + 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f + 6 i \, a d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac {{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x - 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} - 6 i \, a c d^{2} f^{2} x - 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f - 6 i \, a d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+I*a*sinh(f*x+e)),x, algorithm="giac")
Output:
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
Time = 1.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.00 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {\mathrm {cosh}\left (e+f\,x\right )\,\left (a\,c^3\,f^2+6\,a\,c\,d^2\right )\,1{}\mathrm {i}}{f^3}-\frac {\mathrm {sinh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )\,3{}\mathrm {i}}{f^4}+\frac {a\,d^3\,x^4}{4}+a\,c^3\,x+\frac {x\,\mathrm {cosh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )\,3{}\mathrm {i}}{f^3}+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3+\frac {a\,d^3\,x^3\,\mathrm {cosh}\left (e+f\,x\right )\,1{}\mathrm {i}}{f}-\frac {a\,d^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )\,3{}\mathrm {i}}{f^2}-\frac {a\,c\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )\,6{}\mathrm {i}}{f^2}+\frac {a\,c\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )\,3{}\mathrm {i}}{f} \] Input:
int((a + a*sinh(e + f*x)*1i)*(c + d*x)^3,x)
Output:
(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*co sh(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
Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.14 \[ \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx=\frac {a \left (4 \cosh \left (f x +e \right ) c^{3} f^{3} i +12 \cosh \left (f x +e \right ) c^{2} d \,f^{3} i x +12 \cosh \left (f x +e \right ) c \,d^{2} f^{3} i \,x^{2}+24 \cosh \left (f x +e \right ) c \,d^{2} f i +4 \cosh \left (f x +e \right ) d^{3} f^{3} i \,x^{3}+24 \cosh \left (f x +e \right ) d^{3} f i x -12 \sinh \left (f x +e \right ) c^{2} d \,f^{2} i -24 \sinh \left (f x +e \right ) c \,d^{2} f^{2} i x -12 \sinh \left (f x +e \right ) d^{3} f^{2} i \,x^{2}-24 \sinh \left (f x +e \right ) d^{3} i +4 c^{3} f^{4} x +6 c^{2} d \,f^{4} x^{2}+4 c \,d^{2} f^{4} x^{3}+d^{3} f^{4} x^{4}\right )}{4 f^{4}} \] Input:
int((d*x+c)^3*(a+I*a*sinh(f*x+e)),x)
Output:
(a*(4*cosh(e + f*x)*c**3*f**3*i + 12*cosh(e + f*x)*c**2*d*f**3*i*x + 12*co sh(e + f*x)*c*d**2*f**3*i*x**2 + 24*cosh(e + f*x)*c*d**2*f*i + 4*cosh(e + f*x)*d**3*f**3*i*x**3 + 24*cosh(e + f*x)*d**3*f*i*x - 12*sinh(e + f*x)*c** 2*d*f**2*i - 24*sinh(e + f*x)*c*d**2*f**2*i*x - 12*sinh(e + f*x)*d**3*f**2 *i*x**2 - 24*sinh(e + f*x)*d**3*i + 4*c**3*f**4*x + 6*c**2*d*f**4*x**2 + 4 *c*d**2*f**4*x**3 + d**3*f**4*x**4))/(4*f**4)