Integrand size = 20, antiderivative size = 242 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac {b^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}+\frac {a b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \]
-1/2*a^2/d/(d*x+c)^2+b^2*f^2*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/d^3-a*b *f*cosh(f*x+e)/d^2/(d*x+c)+a*b*f^2*cosh(-e+c*f/d)*Shi(c*f/d+f*x)/d^3-b^2*f ^2*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/d^3-a*b*f^2*Chi(c*f/d+f*x)*sinh(- e+c*f/d)/d^3-a*b*sinh(f*x+e)/d/(d*x+c)^2-b^2*f*cosh(f*x+e)*sinh(f*x+e)/d^2 /(d*x+c)-1/2*b^2*sinh(f*x+e)^2/d/(d*x+c)^2
Time = 0.61 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.63 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {-2 a^2 d^2+b^2 d^2-4 a b c d f \cosh (e+f x)-4 a b d^2 f x \cosh (e+f x)-b^2 d^2 \cosh (2 (e+f x))+4 b^2 f^2 (c+d x)^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+4 a b f^2 (c+d x)^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-4 a b d^2 \sinh (e+f x)-2 b^2 c d f \sinh (2 (e+f x))-2 b^2 d^2 f x \sinh (2 (e+f x))+4 a b c^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+8 a b c d f^2 x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 a b d^2 f^2 x^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 b^2 c^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+8 b^2 c d f^2 x \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )}{4 d^3 (c+d x)^2} \]
(-2*a^2*d^2 + b^2*d^2 - 4*a*b*c*d*f*Cosh[e + f*x] - 4*a*b*d^2*f*x*Cosh[e + f*x] - b^2*d^2*Cosh[2*(e + f*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*f^2*(c + d*x)^2*CoshIntegral [f*(c/d + x)]*Sinh[e - (c*f)/d] - 4*a*b*d^2*Sinh[e + f*x] - 2*b^2*c*d*f*Si nh[2*(e + f*x)] - 2*b^2*d^2*f*x*Sinh[2*(e + f*x)] + 4*a*b*c^2*f^2*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 8*a*b*c*d*f^2*x*Cosh[e - (c*f)/d]*Si nhIntegral[f*(c/d + x)] + 4*a*b*d^2*f^2*x^2*Cosh[e - (c*f)/d]*SinhIntegral [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])/(4*d^3*(c + d*x)^2)
Time = 0.72 (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.
| \(\displaystyle \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a-i b \sin (i e+i f x))^2}{(c+d x)^3}dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (\frac {a^2}{(c+d x)^3}+\frac {2 a b \sinh (e+f x)}{(c+d x)^3}+\frac {b^2 \sinh ^2(e+f x)}{(c+d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2}{2 d (c+d x)^2}+\frac {a b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}+\frac {b^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}-\frac {b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}\) |
-1/2*a^2/(d*(c + d*x)^2) - (a*b*f*Cosh[e + f*x])/(d^2*(c + d*x)) + (b^2*f^ 2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 + (a*b*f^2*Co shIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - (a*b*Sinh[e + f*x])/(d* (c + d*x)^2) - (b^2*f*Cosh[e + f*x]*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2* Sinh[e + f*x]^2)/(2*d*(c + d*x)^2) + (a*b*f^2*Cosh[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
3.2.68.3.1 Defintions of rubi rules used
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 = 4.36 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.59
| method | result | size |
| risch | \(-\frac {f^{2} b a \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b a \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b a \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {a^{2}}{2 d \left (d x +c \right )^{2}}+\frac {b^{2}}{4 \left (d x +c \right )^{2} d}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}-\frac {f^{3} a b \,{\mathrm e}^{-f x -e} x}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} a b \,{\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a b \,{\mathrm e}^{-f x -e}}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}\) | \(626\) |
-1/2/d^3*f^2*b*a*exp(f*x+e)/(c*f/d+f*x)^2-1/2/d^3*f^2*b*a*exp(f*x+e)/(c*f/ d+f*x)-1/2/d^3*f^2*b*a*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)-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)
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (242) = 484\).
Time = 0.25 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.44 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} + {\left (2 \, a^{2} - b^{2}\right )} d^{2} + 4 \, {\left (a b d^{2} f x + a b c d f\right )} \cosh \left (f x + e\right ) - 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 4 \, {\left (a b d^{2} + {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
-1/4*(b^2*d^2*cosh(f*x + e)^2 + b^2*d^2*sinh(f*x + e)^2 + (2*a^2 - b^2)*d^ 2 + 4*(a*b*d^2*f*x + a*b*c*d*f)*cosh(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))*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))*c osh(-2*(d*e - c*f)/d) + 4*(a*b*d^2 + (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)*E i((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)
\[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {\left (a + b \sinh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {1}{4} \, b^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + a b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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_integral _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 = 0.28 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 4 \, a b d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 4 \, a b d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 8 \, a b c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 8 \, a b c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 4 \, a b c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 4 \, a b c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} - 2 \, b^{2} d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} f x e^{\left (f x + e\right )} - 4 \, a b d^{2} f x e^{\left (-f x - e\right )} + 2 \, b^{2} d^{2} f x e^{\left (-2 \, f x - 2 \, e\right )} - 2 \, b^{2} c d f e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b c d f e^{\left (f x + e\right )} - 4 \, a b c d f e^{\left (-f x - e\right )} + 2 \, b^{2} c d f e^{\left (-2 \, f x - 2 \, e\right )} - b^{2} d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} e^{\left (f x + e\right )} + 4 \, a b d^{2} e^{\left (-f x - e\right )} - b^{2} d^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 4 \, a^{2} d^{2} + 2 \, b^{2} d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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. \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]