Integrand size = 23, antiderivative size = 174 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {a^2 d^2 x}{4 f^2}+\frac {a^2 (c+d x)^3}{2 d}+\frac {4 i a^2 d^2 \cosh (e+f x)}{f^3}+\frac {2 i a^2 (c+d x)^2 \cosh (e+f x)}{f}-\frac {4 i a^2 d (c+d x) \sinh (e+f x)}{f^2}-\frac {a^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}-\frac {a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {a^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2} \] Output:
1/4*a^2*d^2*x/f^2+1/2*a^2*(d*x+c)^3/d+4*I*a^2*d^2*cosh(f*x+e)/f^3+2*I*a^2* (d*x+c)^2*cosh(f*x+e)/f-4*I*a^2*d*(d*x+c)*sinh(f*x+e)/f^2-1/4*a^2*d^2*cosh (f*x+e)*sinh(f*x+e)/f^3-1/2*a^2*(d*x+c)^2*cosh(f*x+e)*sinh(f*x+e)/f+1/2*a^ 2*d*(d*x+c)*sinh(f*x+e)^2/f^2
Time = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.09 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {a^2 \left (12 c^2 f^3 x+12 c d f^3 x^2+4 d^2 f^3 x^3+16 i \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cosh (e+f x)+2 d f (c+d x) \cosh (2 (e+f x))-32 i c d f \sinh (e+f x)-32 i d^2 f x \sinh (e+f x)-d^2 \sinh (2 (e+f x))-2 c^2 f^2 \sinh (2 (e+f x))-4 c d f^2 x \sinh (2 (e+f x))-2 d^2 f^2 x^2 \sinh (2 (e+f x))\right )}{8 f^3} \] Input:
Integrate[(c + d*x)^2*(a + I*a*Sinh[e + f*x])^2,x]
Output:
(a^2*(12*c^2*f^3*x + 12*c*d*f^3*x^2 + 4*d^2*f^3*x^3 + (16*I)*(c^2*f^2 + 2* c*d*f^2*x + d^2*(2 + f^2*x^2))*Cosh[e + f*x] + 2*d*f*(c + d*x)*Cosh[2*(e + f*x)] - (32*I)*c*d*f*Sinh[e + f*x] - (32*I)*d^2*f*x*Sinh[e + f*x] - d^2*S inh[2*(e + f*x)] - 2*c^2*f^2*Sinh[2*(e + f*x)] - 4*c*d*f^2*x*Sinh[2*(e + f *x)] - 2*d^2*f^2*x^2*Sinh[2*(e + f*x)]))/(8*f^3)
Time = 0.49 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
| \(\displaystyle \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 (a+a \sin (i e+i f x))^2dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (-a^2 (c+d x)^2 \sinh ^2(e+f x)+2 i a^2 (c+d x)^2 \sinh (e+f x)+a^2 (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}-\frac {4 i a^2 d (c+d x) \sinh (e+f x)}{f^2}+\frac {2 i a^2 (c+d x)^2 \cosh (e+f x)}{f}-\frac {a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {a^2 (c+d x)^3}{2 d}+\frac {4 i a^2 d^2 \cosh (e+f x)}{f^3}-\frac {a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {a^2 d^2 x}{4 f^2}\) |
Input:
Int[(c + d*x)^2*(a + I*a*Sinh[e + f*x])^2,x]
Output:
(a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) + ((4*I)*a^2*d^2*Cosh[e + f* x])/f^3 + ((2*I)*a^2*(c + d*x)^2*Cosh[e + f*x])/f - ((4*I)*a^2*d*(c + d*x) *Sinh[e + f*x])/f^2 - (a^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a^2 *(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) + (a^2*d*(c + d*x)*Sinh[e + f*x]^2)/(2*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 = 0.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.81
| method | result | size |
| parallelrisch | \(\frac {2 a^{2} \left (\frac {\left (-\left (d x +c \right )^{2} f^{2}-\frac {d^{2}}{2}\right ) \sinh \left (2 f x +2 e \right )}{8}+\frac {d f \left (d x +c \right ) \cosh \left (2 f x +2 e \right )}{8}+i \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \cosh \left (f x +e \right )-2 i \left (d x +c \right ) f d \sinh \left (f x +e \right )+\frac {3 x \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) f^{3}}{4}+i c^{2} f^{2}-\frac {c d f}{8}+2 i d^{2}\right )}{f^{3}}\) | \(141\) |
| risch | \(\frac {a^{2} d^{2} x^{3}}{2}+\frac {3 a^{2} d c \,x^{2}}{2}+\frac {3 a^{2} c^{2} x}{2}+\frac {a^{2} c^{3}}{2 d}-\frac {a^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-2 d^{2} f x -2 c d f +d^{2}\right ) {\mathrm e}^{2 f x +2 e}}{16 f^{3}}+\frac {i a^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2} f x -2 c d f +2 d^{2}\right ) {\mathrm e}^{f x +e}}{f^{3}}+\frac {i a^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} f x +2 c d f +2 d^{2}\right ) {\mathrm e}^{-f x -e}}{f^{3}}+\frac {a^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{16 f^{3}}\) | \(282\) |
| parts | \(\frac {a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {2 i a^{2} \left (\frac {d^{2} \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d c \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d^{2} e^{2} \cosh \left (f x +e \right )}{f^{2}}-\frac {2 d e c \cosh \left (f x +e \right )}{f}+c^{2} \cosh \left (f x +e \right )\right )}{f}-\frac {a^{2} \left (\frac {d^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \cosh \left (f x +e \right )^{2}}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {2 d^{2} e \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f^{2}}+\frac {d^{2} e^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}+\frac {2 c d \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}-\frac {2 c d e \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )\right )}{f}\) | \(433\) |
| derivativedivides | \(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {2 i d^{2} a^{2} \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 {d^{2} a^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \cosh \left (f x +e \right )^{2}}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {d^{2} e \,a^{2} \left (f x +e \right )^{2}}{f^{2}}+2 i c^{2} a^{2} \cosh \left (f x +e \right )+\frac {2 d^{2} e \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f^{2}}+\frac {d c \,a^{2} \left (f x +e \right )^{2}}{f}-\frac {4 i d e c \,a^{2} \cosh \left (f x +e \right )}{f}-\frac {2 d c \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}+\frac {d^{2} e^{2} a^{2} \left (f x +e \right )}{f^{2}}+\frac {2 i d^{2} e^{2} a^{2} \cosh \left (f x +e \right )}{f^{2}}-\frac {d^{2} e^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}-\frac {2 d e c \,a^{2} \left (f x +e \right )}{f}-\frac {4 i d^{2} e \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d e c \,a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c^{2} a^{2} \left (f x +e \right )+\frac {4 i d c \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-c^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\) | \(550\) |
| default | \(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {2 i d^{2} a^{2} \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 {d^{2} a^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \cosh \left (f x +e \right )^{2}}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {d^{2} e \,a^{2} \left (f x +e \right )^{2}}{f^{2}}+2 i c^{2} a^{2} \cosh \left (f x +e \right )+\frac {2 d^{2} e \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f^{2}}+\frac {d c \,a^{2} \left (f x +e \right )^{2}}{f}-\frac {4 i d e c \,a^{2} \cosh \left (f x +e \right )}{f}-\frac {2 d c \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}+\frac {d^{2} e^{2} a^{2} \left (f x +e \right )}{f^{2}}+\frac {2 i d^{2} e^{2} a^{2} \cosh \left (f x +e \right )}{f^{2}}-\frac {d^{2} e^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}-\frac {2 d e c \,a^{2} \left (f x +e \right )}{f}-\frac {4 i d^{2} e \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d e c \,a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c^{2} a^{2} \left (f x +e \right )+\frac {4 i d c \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-c^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\) | \(550\) |
| orering | \(\text {Expression too large to display}\) | \(1103\) |
Input:
int((d*x+c)^2*(a+I*a*sinh(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
2*a^2/f^3*(1/8*(-(d*x+c)^2*f^2-1/2*d^2)*sinh(2*f*x+2*e)+1/8*d*f*(d*x+c)*co sh(2*f*x+2*e)+I*((d*x+c)^2*f^2+2*d^2)*cosh(f*x+e)-2*I*(d*x+c)*f*d*sinh(f*x +e)+3/4*x*(1/3*x^2*d^2+c*d*x+c^2)*f^3+I*c^2*f^2-1/8*c*d*f+2*I*d^2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (158) = 316\).
Time = 0.10 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.02 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} c d f + a^{2} d^{2} + 2 \, {\left (2 \, a^{2} c d f^{2} + a^{2} d^{2} f\right )} x - {\left (2 \, a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} c d f + a^{2} d^{2} + 2 \, {\left (2 \, a^{2} c d f^{2} - a^{2} d^{2} f\right )} x\right )} e^{\left (4 \, f x + 4 \, e\right )} - 16 \, {\left (-i \, a^{2} d^{2} f^{2} x^{2} - i \, a^{2} c^{2} f^{2} + 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2} + 2 \, {\left (-i \, a^{2} c d f^{2} + i \, a^{2} d^{2} f\right )} x\right )} e^{\left (3 \, f x + 3 \, e\right )} + 8 \, {\left (a^{2} d^{2} f^{3} x^{3} + 3 \, a^{2} c d f^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} x\right )} e^{\left (2 \, f x + 2 \, e\right )} - 16 \, {\left (-i \, a^{2} d^{2} f^{2} x^{2} - i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2} + 2 \, {\left (-i \, a^{2} c d f^{2} - i \, a^{2} d^{2} f\right )} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \] Input:
integrate((d*x+c)^2*(a+I*a*sinh(f*x+e))^2,x, algorithm="fricas")
Output:
1/16*(2*a^2*d^2*f^2*x^2 + 2*a^2*c^2*f^2 + 2*a^2*c*d*f + a^2*d^2 + 2*(2*a^2 *c*d*f^2 + a^2*d^2*f)*x - (2*a^2*d^2*f^2*x^2 + 2*a^2*c^2*f^2 - 2*a^2*c*d*f + a^2*d^2 + 2*(2*a^2*c*d*f^2 - a^2*d^2*f)*x)*e^(4*f*x + 4*e) - 16*(-I*a^2 *d^2*f^2*x^2 - I*a^2*c^2*f^2 + 2*I*a^2*c*d*f - 2*I*a^2*d^2 + 2*(-I*a^2*c*d *f^2 + I*a^2*d^2*f)*x)*e^(3*f*x + 3*e) + 8*(a^2*d^2*f^3*x^3 + 3*a^2*c*d*f^ 3*x^2 + 3*a^2*c^2*f^3*x)*e^(2*f*x + 2*e) - 16*(-I*a^2*d^2*f^2*x^2 - I*a^2* c^2*f^2 - 2*I*a^2*c*d*f - 2*I*a^2*d^2 + 2*(-I*a^2*c*d*f^2 - I*a^2*d^2*f)*x )*e^(f*x + e))*e^(-2*f*x - 2*e)/f^3
Time = 0.48 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.99 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {3 a^{2} c^{2} x}{2} + \frac {3 a^{2} c d x^{2}}{2} + \frac {a^{2} d^{2} x^{3}}{2} + \begin {cases} \frac {\left (\left (32 a^{2} c^{2} f^{11} e^{e} + 64 a^{2} c d f^{11} x e^{e} + 32 a^{2} c d f^{10} e^{e} + 32 a^{2} d^{2} f^{11} x^{2} e^{e} + 32 a^{2} d^{2} f^{10} x e^{e} + 16 a^{2} d^{2} f^{9} e^{e}\right ) e^{- 2 f x} + \left (- 32 a^{2} c^{2} f^{11} e^{5 e} - 64 a^{2} c d f^{11} x e^{5 e} + 32 a^{2} c d f^{10} e^{5 e} - 32 a^{2} d^{2} f^{11} x^{2} e^{5 e} + 32 a^{2} d^{2} f^{10} x e^{5 e} - 16 a^{2} d^{2} f^{9} e^{5 e}\right ) e^{2 f x} + \left (256 i a^{2} c^{2} f^{11} e^{2 e} + 512 i a^{2} c d f^{11} x e^{2 e} + 512 i a^{2} c d f^{10} e^{2 e} + 256 i a^{2} d^{2} f^{11} x^{2} e^{2 e} + 512 i a^{2} d^{2} f^{10} x e^{2 e} + 512 i a^{2} d^{2} f^{9} e^{2 e}\right ) e^{- f x} + \left (256 i a^{2} c^{2} f^{11} e^{4 e} + 512 i a^{2} c d f^{11} x e^{4 e} - 512 i a^{2} c d f^{10} e^{4 e} + 256 i a^{2} d^{2} f^{11} x^{2} e^{4 e} - 512 i a^{2} d^{2} f^{10} x e^{4 e} + 512 i a^{2} d^{2} f^{9} e^{4 e}\right ) e^{f x}\right ) e^{- 3 e}}{256 f^{12}} & \text {for}\: f^{12} e^{3 e} \neq 0 \\\frac {x^{3} \left (- a^{2} d^{2} e^{4 e} + 4 i a^{2} d^{2} e^{3 e} - 4 i a^{2} d^{2} e^{e} - a^{2} d^{2}\right ) e^{- 2 e}}{12} + \frac {x^{2} \left (- a^{2} c d e^{4 e} + 4 i a^{2} c d e^{3 e} - 4 i a^{2} c d e^{e} - a^{2} c d\right ) e^{- 2 e}}{4} + \frac {x \left (- a^{2} c^{2} e^{4 e} + 4 i a^{2} c^{2} e^{3 e} - 4 i a^{2} c^{2} e^{e} - a^{2} c^{2}\right ) e^{- 2 e}}{4} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*(a+I*a*sinh(f*x+e))**2,x)
Output:
3*a**2*c**2*x/2 + 3*a**2*c*d*x**2/2 + a**2*d**2*x**3/2 + Piecewise((((32*a **2*c**2*f**11*exp(e) + 64*a**2*c*d*f**11*x*exp(e) + 32*a**2*c*d*f**10*exp (e) + 32*a**2*d**2*f**11*x**2*exp(e) + 32*a**2*d**2*f**10*x*exp(e) + 16*a* *2*d**2*f**9*exp(e))*exp(-2*f*x) + (-32*a**2*c**2*f**11*exp(5*e) - 64*a**2 *c*d*f**11*x*exp(5*e) + 32*a**2*c*d*f**10*exp(5*e) - 32*a**2*d**2*f**11*x* *2*exp(5*e) + 32*a**2*d**2*f**10*x*exp(5*e) - 16*a**2*d**2*f**9*exp(5*e))* exp(2*f*x) + (256*I*a**2*c**2*f**11*exp(2*e) + 512*I*a**2*c*d*f**11*x*exp( 2*e) + 512*I*a**2*c*d*f**10*exp(2*e) + 256*I*a**2*d**2*f**11*x**2*exp(2*e) + 512*I*a**2*d**2*f**10*x*exp(2*e) + 512*I*a**2*d**2*f**9*exp(2*e))*exp(- f*x) + (256*I*a**2*c**2*f**11*exp(4*e) + 512*I*a**2*c*d*f**11*x*exp(4*e) - 512*I*a**2*c*d*f**10*exp(4*e) + 256*I*a**2*d**2*f**11*x**2*exp(4*e) - 512 *I*a**2*d**2*f**10*x*exp(4*e) + 512*I*a**2*d**2*f**9*exp(4*e))*exp(f*x))*e xp(-3*e)/(256*f**12), Ne(f**12*exp(3*e), 0)), (x**3*(-a**2*d**2*exp(4*e) + 4*I*a**2*d**2*exp(3*e) - 4*I*a**2*d**2*exp(e) - a**2*d**2)*exp(-2*e)/12 + x**2*(-a**2*c*d*exp(4*e) + 4*I*a**2*c*d*exp(3*e) - 4*I*a**2*c*d*exp(e) - a**2*c*d)*exp(-2*e)/4 + x*(-a**2*c**2*exp(4*e) + 4*I*a**2*c**2*exp(3*e) - 4*I*a**2*c**2*exp(e) - a**2*c**2)*exp(-2*e)/4, True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (158) = 316\).
Time = 0.06 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.87 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} - \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} + \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c d + \frac {1}{48} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} + \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} d^{2} + \frac {1}{8} \, a^{2} c^{2} {\left (4 \, x - \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{2} x + 2 i \, a^{2} c d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + i \, a^{2} d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {2 i \, a^{2} c^{2} \cosh \left (f x + e\right )}{f} \] Input:
integrate((d*x+c)^2*(a+I*a*sinh(f*x+e))^2,x, algorithm="maxima")
Output:
1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + 1/8*(4*x^2 - (2*f*x*e^(2*e) - e^(2*e))*e^( 2*f*x)/f^2 + (2*f*x + 1)*e^(-2*f*x - 2*e)/f^2)*a^2*c*d + 1/48*(8*x^3 - 3*( 2*f^2*x^2*e^(2*e) - 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x)/f^3 + 3*(2*f^2*x^2 + 2*f*x + 1)*e^(-2*f*x - 2*e)/f^3)*a^2*d^2 + 1/8*a^2*c^2*(4*x - e^(2*f*x + 2*e)/f + e^(-2*f*x - 2*e)/f) + a^2*c^2*x + 2*I*a^2*c*d*((f*x*e^e - e^e)*e ^(f*x)/f^2 + (f*x + 1)*e^(-f*x - e)/f^2) + I*a^2*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) + 2*I* a^2*c^2*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. 333 vs. \(2 (158) = 316\).
Time = 0.13 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.91 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {1}{2} \, a^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} c d x^{2} + \frac {3}{2} \, a^{2} c^{2} x - \frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac {{\left (i \, a^{2} d^{2} f^{2} x^{2} + 2 i \, a^{2} c d f^{2} x + i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f + 2 i \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} - \frac {{\left (-i \, a^{2} d^{2} f^{2} x^{2} - 2 i \, a^{2} c d f^{2} x - i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} + \frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \] Input:
integrate((d*x+c)^2*(a+I*a*sinh(f*x+e))^2,x, algorithm="giac")
Output:
1/2*a^2*d^2*x^3 + 3/2*a^2*c*d*x^2 + 3/2*a^2*c^2*x - 1/16*(2*a^2*d^2*f^2*x^ 2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 - 2*a^2*d^2*f*x - 2*a^2*c*d*f + a^2*d^ 2)*e^(2*f*x + 2*e)/f^3 + (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 - 2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f + 2*I*a^2*d^2)*e^(f*x + e)/f^3 - ( -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 - 2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f - 2*I*a^2*d^2)*e^(-f*x - e)/f^3 + 1/16*(2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 + 2*a^2*d^2*f*x + 2*a^2*c*d*f + a^2*d^2)* e^(-2*f*x - 2*e)/f^3
Time = 1.65 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.25 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {a^2\,\left (12\,c^2\,x+12\,c\,d\,x^2+4\,d^2\,x^3\right )}{8}+\frac {\frac {a^2\,\left (-d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+d^2\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}+\frac {a^2\,f^2\,\left (-2\,c^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-2\,d^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-4\,c\,d\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+c^2\,\mathrm {cosh}\left (e+f\,x\right )\,16{}\mathrm {i}+d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )\,16{}\mathrm {i}+c\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}-\frac {a^2\,f\,\left (-2\,d^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-2\,c\,d\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )\,32{}\mathrm {i}+c\,d\,\mathrm {sinh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}}{f^3} \] Input:
int((a + a*sinh(e + f*x)*1i)^2*(c + d*x)^2,x)
Output:
(a^2*(12*c^2*x + 4*d^2*x^3 + 12*c*d*x^2))/8 + ((a^2*(d^2*cosh(e + f*x)*32i - d^2*sinh(2*e + 2*f*x)))/8 + (a^2*f^2*(c^2*cosh(e + f*x)*16i - 2*c^2*sin h(2*e + 2*f*x) + d^2*x^2*cosh(e + f*x)*16i - 2*d^2*x^2*sinh(2*e + 2*f*x) + c*d*x*cosh(e + f*x)*32i - 4*c*d*x*sinh(2*e + 2*f*x)))/8 - (a^2*f*(d^2*x*s inh(e + f*x)*32i - 2*d^2*x*cosh(2*e + 2*f*x) + c*d*sinh(e + f*x)*32i - 2*c *d*cosh(2*e + 2*f*x)))/8)/f^3
Time = 0.21 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.47 \[ \int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx=\frac {a^{2} \left (d^{2}+2 c^{2} f^{2}+32 e^{3 f x +3 e} c d \,f^{2} i x +32 e^{f x +e} c d \,f^{2} i x +2 e^{4 f x +4 e} c d f -2 e^{4 f x +4 e} d^{2} f^{2} x^{2}+2 e^{4 f x +4 e} d^{2} f x +16 e^{3 f x +3 e} c^{2} f^{2} i +24 e^{2 f x +2 e} c^{2} f^{3} x +8 e^{2 f x +2 e} d^{2} f^{3} x^{3}+16 e^{f x +e} c^{2} f^{2} i +4 c d \,f^{2} x -4 e^{4 f x +4 e} c d \,f^{2} x -32 e^{3 f x +3 e} c d f i +16 e^{3 f x +3 e} d^{2} f^{2} i \,x^{2}-32 e^{3 f x +3 e} d^{2} f i x +24 e^{2 f x +2 e} c d \,f^{3} x^{2}+32 e^{f x +e} c d f i +16 e^{f x +e} d^{2} f^{2} i \,x^{2}+32 e^{f x +e} d^{2} f i x -2 e^{4 f x +4 e} c^{2} f^{2}+32 e^{3 f x +3 e} d^{2} i +32 e^{f x +e} d^{2} i +2 c d f +2 d^{2} f^{2} x^{2}+2 d^{2} f x -e^{4 f x +4 e} d^{2}\right )}{16 e^{2 f x +2 e} f^{3}} \] Input:
int((d*x+c)^2*(a+I*a*sinh(f*x+e))^2,x)
Output:
(a**2*( - 2*e**(4*e + 4*f*x)*c**2*f**2 - 4*e**(4*e + 4*f*x)*c*d*f**2*x + 2 *e**(4*e + 4*f*x)*c*d*f - 2*e**(4*e + 4*f*x)*d**2*f**2*x**2 + 2*e**(4*e + 4*f*x)*d**2*f*x - e**(4*e + 4*f*x)*d**2 + 16*e**(3*e + 3*f*x)*c**2*f**2*i + 32*e**(3*e + 3*f*x)*c*d*f**2*i*x - 32*e**(3*e + 3*f*x)*c*d*f*i + 16*e**( 3*e + 3*f*x)*d**2*f**2*i*x**2 - 32*e**(3*e + 3*f*x)*d**2*f*i*x + 32*e**(3* e + 3*f*x)*d**2*i + 24*e**(2*e + 2*f*x)*c**2*f**3*x + 24*e**(2*e + 2*f*x)* c*d*f**3*x**2 + 8*e**(2*e + 2*f*x)*d**2*f**3*x**3 + 16*e**(e + f*x)*c**2*f **2*i + 32*e**(e + f*x)*c*d*f**2*i*x + 32*e**(e + f*x)*c*d*f*i + 16*e**(e + f*x)*d**2*f**2*i*x**2 + 32*e**(e + f*x)*d**2*f*i*x + 32*e**(e + f*x)*d** 2*i + 2*c**2*f**2 + 4*c*d*f**2*x + 2*c*d*f + 2*d**2*f**2*x**2 + 2*d**2*f*x + d**2))/(16*e**(2*e + 2*f*x)*f**3)