Integrand size = 20, antiderivative size = 242 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {a^2}{2 d (c+d x)^2}-\frac {a b \cosh (e+f x)}{d (c+d x)^2}-\frac {b^2 \cosh ^2(e+f x)}{2 d (c+d x)^2}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {b^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}-\frac {a b f \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}+\frac {a b f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \] Output:
-1/2*a^2/d/(d*x+c)^2-a*b*cosh(f*x+e)/d/(d*x+c)^2-1/2*b^2*cosh(f*x+e)^2/d/( d*x+c)^2+a*b*f^2*cosh(-e+c*f/d)*Chi(c*f/d+f*x)/d^3+b^2*f^2*cosh(-2*e+2*c*f /d)*Chi(2*c*f/d+2*f*x)/d^3-a*b*f*sinh(f*x+e)/d^2/(d*x+c)-b^2*f*cosh(f*x+e) *sinh(f*x+e)/d^2/(d*x+c)-a*b*f^2*sinh(-e+c*f/d)*Shi(c*f/d+f*x)/d^3-b^2*f^2 *sinh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/d^3
Time = 0.79 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.63 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {2 a^2 d^2+b^2 d^2+4 a b d^2 \cosh (e+f x)+b^2 d^2 \cosh (2 (e+f x))-4 a b f^2 (c+d x)^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-4 b^2 f^2 (c+d x)^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+4 a b c d f \sinh (e+f x)+4 a b d^2 f x \sinh (e+f x)+2 b^2 c d f \sinh (2 (e+f x))+2 b^2 d^2 f x \sinh (2 (e+f x))-4 a b c^2 f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-8 a b c d f^2 x \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-4 a b d^2 f^2 x^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-4 b^2 c^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-8 b^2 c d f^2 x \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )}{4 d^3 (c+d x)^2} \] Input:
Integrate[(a + b*Cosh[e + f*x])^2/(c + d*x)^3,x]
Output:
-1/4*(2*a^2*d^2 + b^2*d^2 + 4*a*b*d^2*Cosh[e + f*x] + b^2*d^2*Cosh[2*(e + f*x)] - 4*a*b*f^2*(c + d*x)^2*Cosh[e - (c*f)/d]*CoshIntegral[f*(c/d + x)] - 4*b^2*f^2*(c + d*x)^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x)) /d] + 4*a*b*c*d*f*Sinh[e + f*x] + 4*a*b*d^2*f*x*Sinh[e + f*x] + 2*b^2*c*d* f*Sinh[2*(e + f*x)] + 2*b^2*d^2*f*x*Sinh[2*(e + f*x)] - 4*a*b*c^2*f^2*Sinh [e - (c*f)/d]*SinhIntegral[f*(c/d + x)] - 8*a*b*c*d*f^2*x*Sinh[e - (c*f)/d ]*SinhIntegral[f*(c/d + x)] - 4*a*b*d^2*f^2*x^2*Sinh[e - (c*f)/d]*SinhInte gral[f*(c/d + x)] - 4*b^2*c^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f* (c + d*x))/d] - 8*b^2*c*d*f^2*x*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] - 4*b^2*d^2*f^2*x^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d])/(d^3*(c + d*x)^2)
Time = 0.73 (sec) , antiderivative size = 242, 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.
| \(\displaystyle \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}{(c+d x)^3}dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (\frac {a^2}{(c+d x)^3}+\frac {2 a b \cosh (e+f x)}{(c+d x)^3}+\frac {b^2 \cosh ^2(e+f x)}{(c+d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2}{2 d (c+d x)^2}+\frac {a b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^3}+\frac {a b f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {a b f \sinh (e+f x)}{d^2 (c+d x)}-\frac {a b \cosh (e+f x)}{d (c+d x)^2}+\frac {b^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}-\frac {b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{2 d (c+d x)^2}\) |
Input:
Int[(a + b*Cosh[e + f*x])^2/(c + d*x)^3,x]
Output:
-1/2*a^2/(d*(c + d*x)^2) - (a*b*Cosh[e + f*x])/(d*(c + d*x)^2) - (b^2*Cosh [e + f*x]^2)/(2*d*(c + d*x)^2) + (a*b*f^2*Cosh[e - (c*f)/d]*CoshIntegral[( c*f)/d + f*x])/d^3 + (b^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 - (a*b*f*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2*f*Cosh[e + f *x]*Sinh[e + f*x])/(d^2*(c + d*x)) + (a*b*f^2*Sinh[e - (c*f)/d]*SinhIntegr al[(c*f)/d + f*x])/d^3 + (b^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c* f)/d + 2*f*x])/d^3
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])
Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(242)=484\).
Time = 2.73 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.59
| method | result | size |
| risch | \(\frac {f^{3} a b \,{\mathrm e}^{-f x -e} x}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} a b \,{\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a b \,{\mathrm e}^{-f x -e}}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {f^{2} a b \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} a b \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} a b \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {a^{2}}{2 d \left (d x +c \right )^{2}}-\frac {b^{2}}{4 \left (d x +c \right )^{2} d}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}\) | \(626\) |
Input:
int((a+b*cosh(f*x+e))^2/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/2*f^3*a*b*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x+1/2*f^3*a*b* exp(-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c-1/2*f^2*a*b*exp(-f*x-e )/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*f^2*a*b/d^3*exp((c*f-d*e)/d)*Ei( 1,f*x+e+(c*f-d*e)/d)-1/2/d^3*f^2*a*b*exp(f*x+e)/(c*f/d+f*x)^2-1/2/d^3*f^2* a*b*exp(f*x+e)/(c*f/d+f*x)-1/2/d^3*f^2*a*b*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-( c*f-d*e)/d)-1/2*a^2/d/(d*x+c)^2-1/4*b^2/(d*x+c)^2/d+1/4*f^3*b^2*exp(-2*f*x -2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x+1/4*f^3*b^2*exp(-2*f*x-2*e)/d^ 2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c-1/8*f^2*b^2*exp(-2*f*x-2*e)/d/(d^2*f ^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*f^2*b^2/d^3*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+ 2*e+2*(c*f-d*e)/d)-1/8*f^2*b^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2-1/4*f^2*b^ 2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)-1/2*f^2*b^2/d^3*exp(-2*(c*f-d*e)/d)*Ei(1, -2*f*x-2*e-2*(c*f-d*e)/d)
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (242) = 484\).
Time = 0.12 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.42 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} + 4 \, a b d^{2} \cosh \left (f x + e\right ) + {\left (2 \, a^{2} + b^{2}\right )} d^{2} - 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 4 \, {\left (a b d^{2} f x + a b c d f + {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+b*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")
Output:
-1/4*(b^2*d^2*cosh(f*x + e)^2 + b^2*d^2*sinh(f*x + e)^2 + 4*a*b*d^2*cosh(f *x + e) + (2*a^2 + b^2)*d^2 - 2*((a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b* c^2*f^2)*Ei((d*f*x + c*f)/d) + (a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^ 2*f^2)*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - 2*((b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d) + (b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d))*cosh(-2*(d*e - c*f) /d) + 4*(a*b*d^2*f*x + a*b*c*d*f + (b^2*d^2*f*x + b^2*c*d*f)*cosh(f*x + e) )*sinh(f*x + e) + 2*((a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^2)*Ei( (d*f*x + c*f)/d) - (a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^2)*Ei(-( d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + 2*((b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^ 2*x + b^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d) - (b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^ 2*x + b^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d))*sinh(-2*(d*e - c*f)/d))/(d^5*x^ 2 + 2*c*d^4*x + c^2*d^3)
\[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {\left (a + b \cosh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \] Input:
integrate((a+b*cosh(f*x+e))**2/(d*x+c)**3,x)
Output:
Integral((a + b*cosh(e + f*x))**2/(c + d*x)**3, x)
Time = 0.13 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} + \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - a b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:
integrate((a+b*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")
Output:
-1/4*b^2*(1/(d^3*x^2 + 2*c*d^2*x + c^2*d) + e^(-2*e + 2*c*f/d)*exp_integra l_e(3, 2*(d*x + c)*f/d)/((d*x + c)^2*d) + e^(2*e - 2*c*f/d)*exp_integral_e (3, -2*(d*x + c)*f/d)/((d*x + c)^2*d)) - a*b*(e^(-e + c*f/d)*exp_integral_ e(3, (d*x + c)*f/d)/((d*x + c)^2*d) + e^(e - c*f/d)*exp_integral_e(3, -(d* x + c)*f/d)/((d*x + c)^2*d)) - 1/2*a^2/(d^3*x^2 + 2*c*d^2*x + c^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (242) = 484\).
Time = 0.13 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+b*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="giac")
Output:
1/8*(4*b^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 4*a*b*d^2 *f^2*x^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*a*b*d^2*f^2*x^2*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 4*b^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(-2* e + 2*c*f/d) + 8*b^2*c*d*f^2*x*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 8 *a*b*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 8*a*b*c*d*f^2*x*Ei(-(d* f*x + c*f)/d)*e^(-e + c*f/d) + 8*b^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(- 2*e + 2*c*f/d) + 4*b^2*c^2*f^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 4 *a*b*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*a*b*c^2*f^2*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 4*b^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(-2*e + 2*c*f/d) - 2*b^2*d^2*f*x*e^(2*f*x + 2*e) - 4*a*b*d^2*f*x*e^(f*x + e) + 4*a *b*d^2*f*x*e^(-f*x - e) + 2*b^2*d^2*f*x*e^(-2*f*x - 2*e) - 2*b^2*c*d*f*e^( 2*f*x + 2*e) - 4*a*b*c*d*f*e^(f*x + e) + 4*a*b*c*d*f*e^(-f*x - e) + 2*b^2* c*d*f*e^(-2*f*x - 2*e) - b^2*d^2*e^(2*f*x + 2*e) - 4*a*b*d^2*e^(f*x + e) - 4*a*b*d^2*e^(-f*x - e) - b^2*d^2*e^(-2*f*x - 2*e) - 4*a^2*d^2 - 2*b^2*d^2 )/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + b*cosh(e + f*x))^2/(c + d*x)^3,x)
Output:
int((a + b*cosh(e + f*x))^2/(c + d*x)^3, x)
\[ \int \frac {(a+b \cosh (e+f x))^2}{(c+d x)^3} \, dx=\frac {e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c^{2} d +2 e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c \,d^{2} x +e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} d^{3} x^{2}+4 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) a b \,c^{2} d +8 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) a b c \,d^{2} x +4 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) a b \,d^{3} x^{2}+e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) b^{2} c^{2} d +2 e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) b^{2} c \,d^{2} x +e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) b^{2} d^{3} x^{2}-2 e^{e} a^{2}-e^{e} b^{2}+4 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) a b \,c^{2} d +8 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) a b c \,d^{2} x +4 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) a b \,d^{3} x^{2}}{4 e^{e} d \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((a+b*cosh(f*x+e))^2/(d*x+c)^3,x)
Output:
(e**(3*e)*int(e**(2*f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x )*b**2*c**2*d + 2*e**(3*e)*int(e**(2*f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x* *2 + d**3*x**3),x)*b**2*c*d**2*x + e**(3*e)*int(e**(2*f*x)/(c**3 + 3*c**2* d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**2*d**3*x**2 + 4*e**(2*e)*int(e**(f* x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b*c**2*d + 8*e**(2 *e)*int(e**(f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b*c* d**2*x + 4*e**(2*e)*int(e**(f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3 *x**3),x)*a*b*d**3*x**2 + e**e*int(1/(e**(2*e + 2*f*x)*c**3 + 3*e**(2*e + 2*f*x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d**2*x**2 + e**(2*e + 2*f*x)*d**3*x **3),x)*b**2*c**2*d + 2*e**e*int(1/(e**(2*e + 2*f*x)*c**3 + 3*e**(2*e + 2* f*x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d**2*x**2 + e**(2*e + 2*f*x)*d**3*x** 3),x)*b**2*c*d**2*x + e**e*int(1/(e**(2*e + 2*f*x)*c**3 + 3*e**(2*e + 2*f* x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d**2*x**2 + e**(2*e + 2*f*x)*d**3*x**3) ,x)*b**2*d**3*x**2 - 2*e**e*a**2 - e**e*b**2 + 4*int(1/(e**(f*x)*c**3 + 3* e**(f*x)*c**2*d*x + 3*e**(f*x)*c*d**2*x**2 + e**(f*x)*d**3*x**3),x)*a*b*c* *2*d + 8*int(1/(e**(f*x)*c**3 + 3*e**(f*x)*c**2*d*x + 3*e**(f*x)*c*d**2*x* *2 + e**(f*x)*d**3*x**3),x)*a*b*c*d**2*x + 4*int(1/(e**(f*x)*c**3 + 3*e**( f*x)*c**2*d*x + 3*e**(f*x)*c*d**2*x**2 + e**(f*x)*d**3*x**3),x)*a*b*d**3*x **2)/(4*e**e*d*(c**2 + 2*c*d*x + d**2*x**2))