Integrand size = 18, antiderivative size = 113 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {a^2 (c+d x)^2}{2 d}-\frac {b^2 (c+d x)^2}{4 d}+\frac {2 a b (c+d x) \cosh (e+f x)}{f}-\frac {2 a b d \sinh (e+f x)}{f^2}+\frac {b^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {b^2 d \sinh ^2(e+f x)}{4 f^2} \] Output:
1/2*a^2*(d*x+c)^2/d-1/4*b^2*(d*x+c)^2/d+2*a*b*(d*x+c)*cosh(f*x+e)/f-2*a*b* d*sinh(f*x+e)/f^2+1/2*b^2*(d*x+c)*cosh(f*x+e)*sinh(f*x+e)/f-1/4*b^2*d*sinh (f*x+e)^2/f^2
Time = 4.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=-\frac {2 \left (2 a^2-b^2\right ) (e+f x) (-2 c f+d (e-f x))-16 a b f (c+d x) \cosh (e+f x)+b^2 d \cosh (2 (e+f x))+16 a b d \sinh (e+f x)-2 b^2 f (c+d x) \sinh (2 (e+f x))}{8 f^2} \] Input:
Integrate[(c + d*x)*(a + b*Sinh[e + f*x])^2,x]
Output:
-1/8*(2*(2*a^2 - b^2)*(e + f*x)*(-2*c*f + d*(e - f*x)) - 16*a*b*f*(c + d*x )*Cosh[e + f*x] + b^2*d*Cosh[2*(e + f*x)] + 16*a*b*d*Sinh[e + f*x] - 2*b^2 *f*(c + d*x)*Sinh[2*(e + f*x)])/f^2
Time = 0.33 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int (c+d x) (a+b \sinh (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x) (a-i b \sin (i e+i f x))^2dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a^2 (c+d x)+2 a b (c+d x) \sinh (e+f x)+b^2 (c+d x) \sinh ^2(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 (c+d x)^2}{2 d}+\frac {2 a b (c+d x) \cosh (e+f x)}{f}-\frac {2 a b d \sinh (e+f x)}{f^2}+\frac {b^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac {b^2 (c+d x)^2}{4 d}-\frac {b^2 d \sinh ^2(e+f x)}{4 f^2}\) |
Input:
Int[(c + d*x)*(a + b*Sinh[e + f*x])^2,x]
Output:
(a^2*(c + d*x)^2)/(2*d) - (b^2*(c + d*x)^2)/(4*d) + (2*a*b*(c + d*x)*Cosh[ e + f*x])/f - (2*a*b*d*Sinh[e + f*x])/f^2 + (b^2*(c + d*x)*Cosh[e + f*x]*S inh[e + f*x])/(2*f) - (b^2*d*Sinh[e + f*x]^2)/(4*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 = 0.56 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98
| method | result | size |
| parallelrisch | \(\frac {2 b^{2} f \left (d x +c \right ) \sinh \left (2 f x +2 e \right )-b^{2} d \cosh \left (2 f x +2 e \right )+16 a b f \left (d x +c \right ) \cosh \left (f x +e \right )-16 d a b \sinh \left (f x +e \right )+\left (\left (-2 d \,x^{2}-4 c x \right ) f^{2}+d \right ) b^{2}+16 c a b f +8 a^{2} \left (\frac {d x}{2}+c \right ) f^{2} x}{8 f^{2}}\) | \(111\) |
| risch | \(\frac {a^{2} d \,x^{2}}{2}+a^{2} c x -\frac {b^{2} d \,x^{2}}{4}-\frac {b^{2} c x}{2}+\frac {b^{2} \left (2 d x f +2 c f -d \right ) {\mathrm e}^{2 f x +2 e}}{16 f^{2}}+\frac {a b \left (d x f +c f -d \right ) {\mathrm e}^{f x +e}}{f^{2}}+\frac {a b \left (d x f +c f +d \right ) {\mathrm e}^{-f x -e}}{f^{2}}-\frac {b^{2} \left (2 d x f +2 c f +d \right ) {\mathrm e}^{-2 f x -2 e}}{16 f^{2}}\) | \(138\) |
| parts | \(a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {b^{2} \left (\frac {d \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}-\frac {d e \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )\right )}{f}+\frac {2 a b \left (\frac {d \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e \cosh \left (f x +e \right )}{f}+c \cosh \left (f x +e \right )\right )}{f}\) | \(176\) |
| derivativedivides | \(\frac {\frac {d \,a^{2} \left (f x +e \right )^{2}}{2 f}+\frac {2 d a b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d \,b^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}-\frac {d e \,a^{2} \left (f x +e \right )}{f}-\frac {2 d e a b \cosh \left (f x +e \right )}{f}-\frac {d e \,b^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c \,a^{2} \left (f x +e \right )+2 c a b \cosh \left (f x +e \right )+c \,b^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\) | \(208\) |
| default | \(\frac {\frac {d \,a^{2} \left (f x +e \right )^{2}}{2 f}+\frac {2 d a b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d \,b^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\cosh \left (f x +e \right )^{2}}{4}\right )}{f}-\frac {d e \,a^{2} \left (f x +e \right )}{f}-\frac {2 d e a b \cosh \left (f x +e \right )}{f}-\frac {d e \,b^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+c \,a^{2} \left (f x +e \right )+2 c a b \cosh \left (f x +e \right )+c \,b^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\) | \(208\) |
| orering | \(\frac {\left (2 d^{5} f^{4} x^{6}+12 c \,d^{4} f^{4} x^{5}+28 c^{2} d^{3} f^{4} x^{4}+32 c^{3} d^{2} f^{4} x^{3}+18 c^{4} d \,f^{4} x^{2}-15 d^{5} f^{2} x^{4}+4 c^{5} f^{4} x -60 c \,d^{4} f^{2} x^{3}-85 c^{2} d^{3} f^{2} x^{2}-50 c^{3} d^{2} f^{2} x -10 c^{4} d \,f^{2}+20 d^{5} x^{2}+40 c \,d^{4} x +8 c^{2} d^{3}\right ) \left (a +b \sinh \left (f x +e \right )\right )^{2}}{4 f^{4} \left (d x +c \right )^{4}}+\frac {\left (10 d^{4} f^{2} x^{4}+40 c \,d^{3} f^{2} x^{3}+55 c^{2} d^{2} f^{2} x^{2}+30 c^{3} d \,f^{2} x +5 c^{4} f^{2}-20 d^{4} x^{2}-40 c \,d^{3} x -8 d^{2} c^{2}\right ) \left (d \left (a +b \sinh \left (f x +e \right )\right )^{2}+2 \left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right ) b f \cosh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{4}}-\frac {\left (5 d^{3} f^{2} x^{4}+20 c \,d^{2} f^{2} x^{3}+25 c^{2} d \,f^{2} x^{2}+10 c^{3} f^{2} x -20 d^{3} x^{2}-40 c \,d^{2} x -8 c^{2} d \right ) \left (4 d \left (a +b \sinh \left (f x +e \right )\right ) b f \cosh \left (f x +e \right )+2 \left (d x +c \right ) b^{2} f^{2} \cosh \left (f x +e \right )^{2}+2 \left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right ) b \,f^{2} \sinh \left (f x +e \right )\right )}{8 f^{4} \left (d x +c \right )^{3}}-\frac {\left (3 x^{2} d^{2}+6 c d x +c^{2}\right ) \left (6 d \,b^{2} f^{2} \cosh \left (f x +e \right )^{2}+6 d \left (a +b \sinh \left (f x +e \right )\right ) b \,f^{2} \sinh \left (f x +e \right )+6 \cosh \left (f x +e \right ) \sinh \left (f x +e \right ) b^{2} f^{3} \left (d x +c \right )+2 \left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right ) b \,f^{3} \cosh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{2}}+\frac {x \left (d x +2 c \right ) \left (24 d \,b^{2} f^{3} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )+8 d \left (a +b \sinh \left (f x +e \right )\right ) b \,f^{3} \cosh \left (f x +e \right )+6 f^{4} \sinh \left (f x +e \right )^{2} b^{2} \left (d x +c \right )+8 \cosh \left (f x +e \right )^{2} f^{4} b^{2} \left (d x +c \right )+2 \left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right ) b \,f^{4} \sinh \left (f x +e \right )\right )}{8 f^{4} \left (d x +c \right )}\) | \(715\) |
Input:
int((d*x+c)*(a+b*sinh(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
1/8*(2*b^2*f*(d*x+c)*sinh(2*f*x+2*e)-b^2*d*cosh(2*f*x+2*e)+16*a*b*f*(d*x+c )*cosh(f*x+e)-16*d*a*b*sinh(f*x+e)+((-2*d*x^2-4*c*x)*f^2+d)*b^2+16*c*a*b*f +8*a^2*(1/2*d*x+c)*f^2*x)/f^2
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {2 \, {\left (2 \, a^{2} - b^{2}\right )} d f^{2} x^{2} + 4 \, {\left (2 \, a^{2} - b^{2}\right )} c f^{2} x - b^{2} d \cosh \left (f x + e\right )^{2} - b^{2} d \sinh \left (f x + e\right )^{2} + 16 \, {\left (a b d f x + a b c f\right )} \cosh \left (f x + e\right ) - 4 \, {\left (4 \, a b d - {\left (b^{2} d f x + b^{2} c f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{8 \, f^{2}} \] Input:
integrate((d*x+c)*(a+b*sinh(f*x+e))^2,x, algorithm="fricas")
Output:
1/8*(2*(2*a^2 - b^2)*d*f^2*x^2 + 4*(2*a^2 - b^2)*c*f^2*x - b^2*d*cosh(f*x + e)^2 - b^2*d*sinh(f*x + e)^2 + 16*(a*b*d*f*x + a*b*c*f)*cosh(f*x + e) - 4*(4*a*b*d - (b^2*d*f*x + b^2*c*f)*cosh(f*x + e))*sinh(f*x + e))/f^2
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (105) = 210\).
Time = 0.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.94 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\begin {cases} a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {2 a b c \cosh {\left (e + f x \right )}}{f} + \frac {2 a b d x \cosh {\left (e + f x \right )}}{f} - \frac {2 a b d \sinh {\left (e + f x \right )}}{f^{2}} + \frac {b^{2} c x \sinh ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {b^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} - \frac {b^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac {b^{2} d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {b^{2} d \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\left (e \right )}\right )^{2} \left (c x + \frac {d x^{2}}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)*(a+b*sinh(f*x+e))**2,x)
Output:
Piecewise((a**2*c*x + a**2*d*x**2/2 + 2*a*b*c*cosh(e + f*x)/f + 2*a*b*d*x* cosh(e + f*x)/f - 2*a*b*d*sinh(e + f*x)/f**2 + b**2*c*x*sinh(e + f*x)**2/2 - b**2*c*x*cosh(e + f*x)**2/2 + b**2*c*sinh(e + f*x)*cosh(e + f*x)/(2*f) + b**2*d*x**2*sinh(e + f*x)**2/4 - b**2*d*x**2*cosh(e + f*x)**2/4 + b**2*d *x*sinh(e + f*x)*cosh(e + f*x)/(2*f) - b**2*d*sinh(e + f*x)**2/(4*f**2), N e(f, 0)), ((a + b*sinh(e))**2*(c*x + d*x**2/2), True))
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.45 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {1}{2} \, a^{2} d x^{2} - \frac {1}{16} \, {\left (4 \, x^{2} - \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} + \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} b^{2} d - \frac {1}{8} \, b^{2} c {\left (4 \, x - \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c x + a b d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {2 \, a b c \cosh \left (f x + e\right )}{f} \] Input:
integrate((d*x+c)*(a+b*sinh(f*x+e))^2,x, algorithm="maxima")
Output:
1/2*a^2*d*x^2 - 1/16*(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*d - 1/8*b^2*c*(4*x - e^(2*f*x + 2*e)/f + e^(-2*f*x - 2*e)/f) + a^2*c*x + a*b*d*((f*x*e^e - e^e)*e^(f*x)/f^2 + (f *x + 1)*e^(-f*x - e)/f^2) + 2*a*b*c*cosh(f*x + e)/f
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.41 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {1}{2} \, a^{2} d x^{2} - \frac {1}{4} \, b^{2} d x^{2} + a^{2} c x - \frac {1}{2} \, b^{2} c x + \frac {{\left (2 \, b^{2} d f x + 2 \, b^{2} c f - b^{2} d\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{2}} + \frac {{\left (a b d f x + a b c f - a b d\right )} e^{\left (f x + e\right )}}{f^{2}} + \frac {{\left (a b d f x + a b c f + a b d\right )} e^{\left (-f x - e\right )}}{f^{2}} - \frac {{\left (2 \, b^{2} d f x + 2 \, b^{2} c f + b^{2} d\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{2}} \] Input:
integrate((d*x+c)*(a+b*sinh(f*x+e))^2,x, algorithm="giac")
Output:
1/2*a^2*d*x^2 - 1/4*b^2*d*x^2 + a^2*c*x - 1/2*b^2*c*x + 1/16*(2*b^2*d*f*x + 2*b^2*c*f - b^2*d)*e^(2*f*x + 2*e)/f^2 + (a*b*d*f*x + a*b*c*f - a*b*d)*e ^(f*x + e)/f^2 + (a*b*d*f*x + a*b*c*f + a*b*d)*e^(-f*x - e)/f^2 - 1/16*(2* b^2*d*f*x + 2*b^2*c*f + b^2*d)*e^(-2*f*x - 2*e)/f^2
Time = 0.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.19 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {a^2\,d\,x^2}{2}-\frac {b^2\,d\,x^2}{4}+a^2\,c\,x-\frac {b^2\,c\,x}{2}-\frac {b^2\,d\,{\mathrm {cosh}\left (e+f\,x\right )}^2}{4\,f^2}+\frac {b^2\,c\,\mathrm {cosh}\left (e+f\,x\right )\,\mathrm {sinh}\left (e+f\,x\right )}{2\,f}+\frac {2\,a\,b\,c\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {2\,a\,b\,d\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {2\,a\,b\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f}+\frac {b^2\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )\,\mathrm {sinh}\left (e+f\,x\right )}{2\,f} \] Input:
int((a + b*sinh(e + f*x))^2*(c + d*x),x)
Output:
(a^2*d*x^2)/2 - (b^2*d*x^2)/4 + a^2*c*x - (b^2*c*x)/2 - (b^2*d*cosh(e + f* x)^2)/(4*f^2) + (b^2*c*cosh(e + f*x)*sinh(e + f*x))/(2*f) + (2*a*b*c*cosh( e + f*x))/f - (2*a*b*d*sinh(e + f*x))/f^2 + (2*a*b*d*x*cosh(e + f*x))/f + (b^2*d*x*cosh(e + f*x)*sinh(e + f*x))/(2*f)
Time = 0.25 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.31 \[ \int (c+d x) (a+b \sinh (e+f x))^2 \, dx=\frac {2 e^{4 f x +4 e} b^{2} c f +2 e^{4 f x +4 e} b^{2} d f x -e^{4 f x +4 e} b^{2} d +16 e^{3 f x +3 e} a b c f +16 e^{3 f x +3 e} a b d f x -16 e^{3 f x +3 e} a b d +16 e^{2 f x +2 e} a^{2} c \,f^{2} x +8 e^{2 f x +2 e} a^{2} d \,f^{2} x^{2}-8 e^{2 f x +2 e} b^{2} c \,f^{2} x -4 e^{2 f x +2 e} b^{2} d \,f^{2} x^{2}+16 e^{f x +e} a b c f +16 e^{f x +e} a b d f x +16 e^{f x +e} a b d -2 b^{2} c f -2 b^{2} d f x -b^{2} d}{16 e^{2 f x +2 e} f^{2}} \] Input:
int((d*x+c)*(a+b*sinh(f*x+e))^2,x)
Output:
(2*e**(4*e + 4*f*x)*b**2*c*f + 2*e**(4*e + 4*f*x)*b**2*d*f*x - e**(4*e + 4 *f*x)*b**2*d + 16*e**(3*e + 3*f*x)*a*b*c*f + 16*e**(3*e + 3*f*x)*a*b*d*f*x - 16*e**(3*e + 3*f*x)*a*b*d + 16*e**(2*e + 2*f*x)*a**2*c*f**2*x + 8*e**(2 *e + 2*f*x)*a**2*d*f**2*x**2 - 8*e**(2*e + 2*f*x)*b**2*c*f**2*x - 4*e**(2* e + 2*f*x)*b**2*d*f**2*x**2 + 16*e**(e + f*x)*a*b*c*f + 16*e**(e + f*x)*a* b*d*f*x + 16*e**(e + f*x)*a*b*d - 2*b**2*c*f - 2*b**2*d*f*x - b**2*d)/(16* e**(2*e + 2*f*x)*f**2)