Integrand size = 20, antiderivative size = 182 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {b^2 d^2 x}{4 f^2}+\frac {a^2 (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{6 d}-\frac {4 a b d (c+d x) \cosh (e+f x)}{f^2}-\frac {b^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac {4 a b d^2 \sinh (e+f x)}{f^3}+\frac {2 a b (c+d x)^2 \sinh (e+f x)}{f}+\frac {b^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f} \] Output:
1/4*b^2*d^2*x/f^2+1/3*a^2*(d*x+c)^3/d+1/6*b^2*(d*x+c)^3/d-4*a*b*d*(d*x+c)* cosh(f*x+e)/f^2-1/2*b^2*d*(d*x+c)*cosh(f*x+e)^2/f^2+4*a*b*d^2*sinh(f*x+e)/ f^3+2*a*b*(d*x+c)^2*sinh(f*x+e)/f+1/4*b^2*d^2*cosh(f*x+e)*sinh(f*x+e)/f^3+ 1/2*b^2*(d*x+c)^2*cosh(f*x+e)*sinh(f*x+e)/f
Time = 0.58 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.38 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {1}{24} \left (24 a^2 c^2 x+12 b^2 c^2 x+24 a^2 c d x^2+12 b^2 c d x^2+8 a^2 d^2 x^3+4 b^2 d^2 x^3-\frac {96 a b d (c+d x) \cosh (e+f x)}{f^2}-\frac {6 b^2 d (c+d x) \cosh (2 (e+f x))}{f^2}+\frac {96 a b d^2 \sinh (e+f x)}{f^3}+\frac {48 a b c^2 \sinh (e+f x)}{f}+\frac {96 a b c d x \sinh (e+f x)}{f}+\frac {48 a b d^2 x^2 \sinh (e+f x)}{f}+\frac {3 b^2 d^2 \sinh (2 (e+f x))}{f^3}+\frac {6 b^2 c^2 \sinh (2 (e+f x))}{f}+\frac {12 b^2 c d x \sinh (2 (e+f x))}{f}+\frac {6 b^2 d^2 x^2 \sinh (2 (e+f x))}{f}\right ) \] Input:
Integrate[(c + d*x)^2*(a + b*Cosh[e + f*x])^2,x]
Output:
(24*a^2*c^2*x + 12*b^2*c^2*x + 24*a^2*c*d*x^2 + 12*b^2*c*d*x^2 + 8*a^2*d^2 *x^3 + 4*b^2*d^2*x^3 - (96*a*b*d*(c + d*x)*Cosh[e + f*x])/f^2 - (6*b^2*d*( c + d*x)*Cosh[2*(e + f*x)])/f^2 + (96*a*b*d^2*Sinh[e + f*x])/f^3 + (48*a*b *c^2*Sinh[e + f*x])/f + (96*a*b*c*d*x*Sinh[e + f*x])/f + (48*a*b*d^2*x^2*S inh[e + f*x])/f + (3*b^2*d^2*Sinh[2*(e + f*x)])/f^3 + (6*b^2*c^2*Sinh[2*(e + f*x)])/f + (12*b^2*c*d*x*Sinh[2*(e + f*x)])/f + (6*b^2*d^2*x^2*Sinh[2*( e + f*x)])/f)/24
Time = 0.44 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
| \(\displaystyle \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 \left (a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \cosh (e+f x)+b^2 (c+d x)^2 \cosh ^2(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d (c+d x) \cosh (e+f x)}{f^2}+\frac {2 a b (c+d x)^2 \sinh (e+f x)}{f}+\frac {4 a b d^2 \sinh (e+f x)}{f^3}-\frac {b^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac {b^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {b^2 (c+d x)^3}{6 d}+\frac {b^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {b^2 d^2 x}{4 f^2}\) |
Input:
Int[(c + d*x)^2*(a + b*Cosh[e + f*x])^2,x]
Output:
(b^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(6*d) - (4*a*b*d*(c + d*x)*Cosh[e + f*x])/f^2 - (b^2*d*(c + d*x)*Cosh[e + f*x]^2)/ (2*f^2) + (4*a*b*d^2*Sinh[e + f*x])/f^3 + (2*a*b*(c + d*x)^2*Sinh[e + f*x] )/f + (b^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (b^2*(c + d*x)^2*Cos h[e + f*x]*Sinh[e + f*x])/(2*f)
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 = 1.57 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.80
| method | result | size |
| parallelrisch | \(\frac {b^{2} \left (\left (d x +c \right )^{2} f^{2}+\frac {d^{2}}{2}\right ) \sinh \left (2 f x +2 e \right )-b^{2} d f \left (d x +c \right ) \cosh \left (2 f x +2 e \right )+8 b a \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \sinh \left (f x +e \right )+4 \left (-4 b a d \left (d x +c \right ) \cosh \left (f x +e \right )+\left (a^{2}+\frac {b^{2}}{2}\right ) \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) x \,f^{2}-4 c \left (a -\frac {b}{16}\right ) b d \right ) f}{4 f^{3}}\) | \(145\) |
| risch | \(\frac {d^{2} a^{2} x^{3}}{3}+\frac {d^{2} b^{2} x^{3}}{6}+a^{2} d c \,x^{2}+\frac {d \,b^{2} c \,x^{2}}{2}+a^{2} c^{2} x +\frac {b^{2} c^{2} x}{2}+\frac {a^{2} c^{3}}{3 d}+\frac {b^{2} c^{3}}{6 d}+\frac {b^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-2 d^{2} f x -2 c d f +d^{2}\right ) {\mathrm e}^{2 f x +2 e}}{16 f^{3}}+\frac {a b \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2} f x -2 c d f +2 d^{2}\right ) {\mathrm e}^{f x +e}}{f^{3}}-\frac {a b \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} f x +2 c d f +2 d^{2}\right ) {\mathrm e}^{-f x -e}}{f^{3}}-\frac {b^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{16 f^{3}}\) | \(316\) |
| parts | \(\frac {a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} \left (\frac {d^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \cosh \left (f x +e \right )^{2}}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {2 d^{2} e \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f^{2}}+\frac {d^{2} e^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {2 c d \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}-\frac {2 c d e \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+c^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 a b \left (\frac {d^{2} \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}+\frac {d^{2} e^{2} \sinh \left (f x +e \right )}{f^{2}}+\frac {2 c d \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}-\frac {2 d e c \sinh \left (f x +e \right )}{f}+c^{2} \sinh \left (f x +e \right )\right )}{f}\) | \(430\) |
| derivativedivides | \(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {2 d^{2} a b \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d^{2} b^{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}}-\frac {4 d^{2} e a b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \,b^{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 d c a b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}+\frac {2 d c \,b^{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 d^{2} e^{2} a b \sinh \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} b^{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 d e c a b \sinh \left (f x +e \right )}{f}-\frac {2 d e c \,b^{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 )+2 c^{2} a b \sinh \left (f x +e \right )+b^{2} 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 )}{f}\) | \(535\) |
| default | \(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {2 d^{2} a b \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d^{2} b^{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}}-\frac {4 d^{2} e a b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \,b^{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 d c a b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}+\frac {2 d c \,b^{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 d^{2} e^{2} a b \sinh \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} b^{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 d e c a b \sinh \left (f x +e \right )}{f}-\frac {2 d e c \,b^{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 )+2 c^{2} a b \sinh \left (f x +e \right )+b^{2} 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 )}{f}\) | \(535\) |
| orering | \(\text {Expression too large to display}\) | \(1070\) |
Input:
int((d*x+c)^2*(a+b*cosh(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
1/4*(b^2*((d*x+c)^2*f^2+1/2*d^2)*sinh(2*f*x+2*e)-b^2*d*f*(d*x+c)*cosh(2*f* x+2*e)+8*b*a*((d*x+c)^2*f^2+2*d^2)*sinh(f*x+e)+4*(-4*b*a*d*(d*x+c)*cosh(f* x+e)+(a^2+1/2*b^2)*(1/3*x^2*d^2+c*d*x+c^2)*x*f^2-4*c*(a-1/16*b)*b*d)*f)/f^ 3
Time = 0.10 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.32 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {2 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} f^{3} x^{3} + 6 \, {\left (2 \, a^{2} + b^{2}\right )} c d f^{3} x^{2} + 6 \, {\left (2 \, a^{2} + b^{2}\right )} c^{2} f^{3} x - 3 \, {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )^{2} - 3 \, {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \sinh \left (f x + e\right )^{2} - 48 \, {\left (a b d^{2} f x + a b c d f\right )} \cosh \left (f x + e\right ) + 3 \, {\left (8 \, a b d^{2} f^{2} x^{2} + 16 \, a b c d f^{2} x + 8 \, a b c^{2} f^{2} + 16 \, a b d^{2} + {\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} + b^{2} d^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{12 \, f^{3}} \] Input:
integrate((d*x+c)^2*(a+b*cosh(f*x+e))^2,x, algorithm="fricas")
Output:
1/12*(2*(2*a^2 + b^2)*d^2*f^3*x^3 + 6*(2*a^2 + b^2)*c*d*f^3*x^2 + 6*(2*a^2 + b^2)*c^2*f^3*x - 3*(b^2*d^2*f*x + b^2*c*d*f)*cosh(f*x + e)^2 - 3*(b^2*d ^2*f*x + b^2*c*d*f)*sinh(f*x + e)^2 - 48*(a*b*d^2*f*x + a*b*c*d*f)*cosh(f* x + e) + 3*(8*a*b*d^2*f^2*x^2 + 16*a*b*c*d*f^2*x + 8*a*b*c^2*f^2 + 16*a*b* d^2 + (2*b^2*d^2*f^2*x^2 + 4*b^2*c*d*f^2*x + 2*b^2*c^2*f^2 + b^2*d^2)*cosh (f*x + e))*sinh(f*x + e))/f^3
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (177) = 354\).
Time = 0.32 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.51 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\begin {cases} a^{2} c^{2} x + a^{2} c d x^{2} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {2 a b c^{2} \sinh {\left (e + f x \right )}}{f} + \frac {4 a b c d x \sinh {\left (e + f x \right )}}{f} - \frac {4 a b c d \cosh {\left (e + f x \right )}}{f^{2}} + \frac {2 a b d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {4 a b d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {4 a b d^{2} \sinh {\left (e + f x \right )}}{f^{3}} - \frac {b^{2} c^{2} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{2} x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {b^{2} c d x^{2} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d x^{2} \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{f} - \frac {b^{2} c d \sinh ^{2}{\left (e + f x \right )}}{2 f^{2}} - \frac {b^{2} d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{6} + \frac {b^{2} d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{6} + \frac {b^{2} d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {b^{2} d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {b^{2} d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {b^{2} d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} & \text {for}\: f \neq 0 \\\left (a + b \cosh {\left (e \right )}\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*(a+b*cosh(f*x+e))**2,x)
Output:
Piecewise((a**2*c**2*x + a**2*c*d*x**2 + a**2*d**2*x**3/3 + 2*a*b*c**2*sin h(e + f*x)/f + 4*a*b*c*d*x*sinh(e + f*x)/f - 4*a*b*c*d*cosh(e + f*x)/f**2 + 2*a*b*d**2*x**2*sinh(e + f*x)/f - 4*a*b*d**2*x*cosh(e + f*x)/f**2 + 4*a* b*d**2*sinh(e + f*x)/f**3 - b**2*c**2*x*sinh(e + f*x)**2/2 + b**2*c**2*x*c osh(e + f*x)**2/2 + b**2*c**2*sinh(e + f*x)*cosh(e + f*x)/(2*f) - b**2*c*d *x**2*sinh(e + f*x)**2/2 + b**2*c*d*x**2*cosh(e + f*x)**2/2 + b**2*c*d*x*s inh(e + f*x)*cosh(e + f*x)/f - b**2*c*d*sinh(e + f*x)**2/(2*f**2) - b**2*d **2*x**3*sinh(e + f*x)**2/6 + b**2*d**2*x**3*cosh(e + f*x)**2/6 + b**2*d** 2*x**2*sinh(e + f*x)*cosh(e + f*x)/(2*f) - b**2*d**2*x*sinh(e + f*x)**2/(4 *f**2) - b**2*d**2*x*cosh(e + f*x)**2/(4*f**2) + b**2*d**2*sinh(e + f*x)*c osh(e + f*x)/(4*f**3), Ne(f, 0)), ((a + b*cosh(e))**2*(c**2*x + c*d*x**2 + d**2*x**3/3), True))
Time = 0.06 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.78 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} + \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} b^{2} c d + \frac {1}{48} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} b^{2} d^{2} + \frac {1}{8} \, b^{2} c^{2} {\left (4 \, x + \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{2} x + 2 \, a b c d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + a b d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {2 \, a b c^{2} \sinh \left (f x + e\right )}{f} \] Input:
integrate((d*x+c)^2*(a+b*cosh(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)*b^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)*b^2*d^2 + 1/8*b^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*a*b*c*d*((f*x*e^e - e^e)*e^( f*x)/f^2 - (f*x + 1)*e^(-f*x - e)/f^2) + a*b*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*a*b*c^ 2*sinh(f*x + e)/f
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (170) = 340\).
Time = 0.12 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.90 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{6} \, b^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac {1}{2} \, b^{2} c d x^{2} + a^{2} c^{2} x + \frac {1}{2} \, b^{2} c^{2} x + \frac {{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} - 2 \, b^{2} d^{2} f x - 2 \, b^{2} c d f + b^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac {{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} - 2 \, a b d^{2} f x - 2 \, a b c d f + 2 \, a b d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} - \frac {{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} + 2 \, a b d^{2} f x + 2 \, a b c d f + 2 \, a b d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} - \frac {{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} + 2 \, b^{2} d^{2} f x + 2 \, b^{2} c d f + b^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \] Input:
integrate((d*x+c)^2*(a+b*cosh(f*x+e))^2,x, algorithm="giac")
Output:
1/3*a^2*d^2*x^3 + 1/6*b^2*d^2*x^3 + a^2*c*d*x^2 + 1/2*b^2*c*d*x^2 + a^2*c^ 2*x + 1/2*b^2*c^2*x + 1/16*(2*b^2*d^2*f^2*x^2 + 4*b^2*c*d*f^2*x + 2*b^2*c^ 2*f^2 - 2*b^2*d^2*f*x - 2*b^2*c*d*f + b^2*d^2)*e^(2*f*x + 2*e)/f^3 + (a*b* d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^2 - 2*a*b*d^2*f*x - 2*a*b*c*d*f + 2*a*b*d^2)*e^(f*x + e)/f^3 - (a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^ 2*f^2 + 2*a*b*d^2*f*x + 2*a*b*c*d*f + 2*a*b*d^2)*e^(-f*x - e)/f^3 - 1/16*( 2*b^2*d^2*f^2*x^2 + 4*b^2*c*d*f^2*x + 2*b^2*c^2*f^2 + 2*b^2*d^2*f*x + 2*b^ 2*c*d*f + b^2*d^2)*e^(-2*f*x - 2*e)/f^3
Time = 2.37 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.54 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=a^2\,c^2\,x+\frac {b^2\,c^2\,x}{2}+\frac {a^2\,d^2\,x^3}{3}+\frac {b^2\,d^2\,x^3}{6}+\frac {b^2\,c^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}+\frac {b^2\,d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{8\,f^3}+a^2\,c\,d\,x^2+\frac {b^2\,c\,d\,x^2}{2}+\frac {2\,a\,b\,c^2\,\mathrm {sinh}\left (e+f\,x\right )}{f}+\frac {4\,a\,b\,d^2\,\mathrm {sinh}\left (e+f\,x\right )}{f^3}+\frac {b^2\,d^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {b^2\,c\,d\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{4\,f^2}-\frac {b^2\,d^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{4\,f^2}-\frac {4\,a\,b\,c\,d\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}-\frac {4\,a\,b\,d^2\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {2\,a\,b\,d^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f}+\frac {b^2\,c\,d\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{2\,f}+\frac {4\,a\,b\,c\,d\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f} \] Input:
int((a + b*cosh(e + f*x))^2*(c + d*x)^2,x)
Output:
a^2*c^2*x + (b^2*c^2*x)/2 + (a^2*d^2*x^3)/3 + (b^2*d^2*x^3)/6 + (b^2*c^2*s inh(2*e + 2*f*x))/(4*f) + (b^2*d^2*sinh(2*e + 2*f*x))/(8*f^3) + a^2*c*d*x^ 2 + (b^2*c*d*x^2)/2 + (2*a*b*c^2*sinh(e + f*x))/f + (4*a*b*d^2*sinh(e + f* x))/f^3 + (b^2*d^2*x^2*sinh(2*e + 2*f*x))/(4*f) - (b^2*c*d*cosh(2*e + 2*f* x))/(4*f^2) - (b^2*d^2*x*cosh(2*e + 2*f*x))/(4*f^2) - (4*a*b*c*d*cosh(e + f*x))/f^2 - (4*a*b*d^2*x*cosh(e + f*x))/f^2 + (2*a*b*d^2*x^2*sinh(e + f*x) )/f + (b^2*c*d*x*sinh(2*e + 2*f*x))/(2*f) + (4*a*b*c*d*x*sinh(e + f*x))/f
Time = 0.15 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.04 \[ \int (c+d x)^2 (a+b \cosh (e+f x))^2 \, dx=\frac {-6 e^{4 f x +4 e} b^{2} c d f +6 e^{4 f x +4 e} b^{2} d^{2} f^{2} x^{2}-6 e^{4 f x +4 e} b^{2} d^{2} f x +48 e^{3 f x +3 e} a b \,c^{2} f^{2}+48 e^{2 f x +2 e} a^{2} c^{2} f^{3} x +16 e^{2 f x +2 e} a^{2} d^{2} f^{3} x^{3}+24 e^{2 f x +2 e} b^{2} c^{2} f^{3} x +8 e^{2 f x +2 e} b^{2} d^{2} f^{3} x^{3}-48 e^{f x +e} a b \,c^{2} f^{2}-12 b^{2} c d \,f^{2} x +6 e^{4 f x +4 e} b^{2} c^{2} f^{2}+96 e^{3 f x +3 e} a b \,d^{2}-96 e^{f x +e} a b \,d^{2}-6 b^{2} c d f -6 b^{2} d^{2} f^{2} x^{2}-6 b^{2} d^{2} f x +3 e^{4 f x +4 e} b^{2} d^{2}-6 b^{2} c^{2} f^{2}+12 e^{4 f x +4 e} b^{2} c d \,f^{2} x -96 e^{3 f x +3 e} a b c d f +48 e^{3 f x +3 e} a b \,d^{2} f^{2} x^{2}-96 e^{3 f x +3 e} a b \,d^{2} f x +48 e^{2 f x +2 e} a^{2} c d \,f^{3} x^{2}+24 e^{2 f x +2 e} b^{2} c d \,f^{3} x^{2}-96 e^{f x +e} a b c d f -48 e^{f x +e} a b \,d^{2} f^{2} x^{2}-96 e^{f x +e} a b \,d^{2} f x -3 b^{2} d^{2}-96 e^{f x +e} a b c d \,f^{2} x +96 e^{3 f x +3 e} a b c d \,f^{2} x}{48 e^{2 f x +2 e} f^{3}} \] Input:
int((d*x+c)^2*(a+b*cosh(f*x+e))^2,x)
Output:
(6*e**(4*e + 4*f*x)*b**2*c**2*f**2 + 12*e**(4*e + 4*f*x)*b**2*c*d*f**2*x - 6*e**(4*e + 4*f*x)*b**2*c*d*f + 6*e**(4*e + 4*f*x)*b**2*d**2*f**2*x**2 - 6*e**(4*e + 4*f*x)*b**2*d**2*f*x + 3*e**(4*e + 4*f*x)*b**2*d**2 + 48*e**(3 *e + 3*f*x)*a*b*c**2*f**2 + 96*e**(3*e + 3*f*x)*a*b*c*d*f**2*x - 96*e**(3* e + 3*f*x)*a*b*c*d*f + 48*e**(3*e + 3*f*x)*a*b*d**2*f**2*x**2 - 96*e**(3*e + 3*f*x)*a*b*d**2*f*x + 96*e**(3*e + 3*f*x)*a*b*d**2 + 48*e**(2*e + 2*f*x )*a**2*c**2*f**3*x + 48*e**(2*e + 2*f*x)*a**2*c*d*f**3*x**2 + 16*e**(2*e + 2*f*x)*a**2*d**2*f**3*x**3 + 24*e**(2*e + 2*f*x)*b**2*c**2*f**3*x + 24*e* *(2*e + 2*f*x)*b**2*c*d*f**3*x**2 + 8*e**(2*e + 2*f*x)*b**2*d**2*f**3*x**3 - 48*e**(e + f*x)*a*b*c**2*f**2 - 96*e**(e + f*x)*a*b*c*d*f**2*x - 96*e** (e + f*x)*a*b*c*d*f - 48*e**(e + f*x)*a*b*d**2*f**2*x**2 - 96*e**(e + f*x) *a*b*d**2*f*x - 96*e**(e + f*x)*a*b*d**2 - 6*b**2*c**2*f**2 - 12*b**2*c*d* f**2*x - 6*b**2*c*d*f - 6*b**2*d**2*f**2*x**2 - 6*b**2*d**2*f*x - 3*b**2*d **2)/(48*e**(2*e + 2*f*x)*f**3)