Integrand size = 20, antiderivative size = 207 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)^2}+\frac {a^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {a^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {f x}{2}\right )}{d^2 (c+d x)}+\frac {a^2 f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {a^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \] Output:
-2*a^2*cosh(1/2*f*x+1/2*e)^4/d/(d*x+c)^2+a^2*f^2*cosh(-e+c*f/d)*Chi(c*f/d+ f*x)/d^3+a^2*f^2*cosh(-2*e+2*c*f/d)*Chi(2*c*f/d+2*f*x)/d^3-4*a^2*f*cosh(1/ 2*f*x+1/2*e)^3*sinh(1/2*f*x+1/2*e)/d^2/(d*x+c)-a^2*f^2*sinh(-e+c*f/d)*Shi( c*f/d+f*x)/d^3-a^2*f^2*sinh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/d^3
Time = 1.08 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.71 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=\frac {a^2 \left (-3 d^2-4 d^2 \cosh (e+f x)-d^2 \cosh (2 (e+f x))+4 f^2 (c+d x)^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )+4 f^2 (c+d x)^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )-4 c d f \sinh (e+f x)-4 d^2 f x \sinh (e+f x)-2 c d f \sinh (2 (e+f x))-2 d^2 f x \sinh (2 (e+f x))+4 c^2 f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+8 c d f^2 x \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 d^2 f^2 x^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 c^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+8 c d f^2 x \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 d^2 f^2 x^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 d^3 (c+d x)^2} \] Input:
Integrate[(a + a*Cosh[e + f*x])^2/(c + d*x)^3,x]
Output:
(a^2*(-3*d^2 - 4*d^2*Cosh[e + f*x] - d^2*Cosh[2*(e + f*x)] + 4*f^2*(c + d* x)^2*Cosh[e - (c*f)/d]*CoshIntegral[f*(c/d + x)] + 4*f^2*(c + d*x)^2*Cosh[ 2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x))/d] - 4*c*d*f*Sinh[e + f*x] - 4*d^2*f*x*Sinh[e + f*x] - 2*c*d*f*Sinh[2*(e + f*x)] - 2*d^2*f*x*Sinh[2*(e + f*x)] + 4*c^2*f^2*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 8*c*d*f ^2*x*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*d^2*f^2*x^2*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*c^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhI ntegral[(2*f*(c + d*x))/d] + 8*c*d*f^2*x*Sinh[2*e - (2*c*f)/d]*SinhIntegra l[(2*f*(c + d*x))/d] + 4*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 = 0.92 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.44, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 3799, 3042, 3795, 3042, 3793, 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.
\(\displaystyle \int \frac {(a \cosh (e+f x)+a)^2}{(c+d x)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+a \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}{(c+d x)^3}dx\) |
\(\Big \downarrow \) 3799 |
\(\displaystyle 4 a^2 \int \frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{(c+d x)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 a^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^4}{(c+d x)^3}dx\) |
\(\Big \downarrow \) 3795 |
\(\displaystyle 4 a^2 \left (\frac {2 f^2 \int \frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{c+d x}dx}{d^2}-\frac {3 f^2 \int \frac {\cosh ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{c+d x}dx}{2 d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 d (c+d x)^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^2}{c+d x}dx}{2 d^2}+\frac {2 f^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^4}{c+d x}dx}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 d (c+d x)^2}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \int \left (\frac {\cosh (e+f x)}{2 (c+d x)}+\frac {1}{2 (c+d x)}\right )dx}{2 d^2}+\frac {2 f^2 \int \left (\frac {\cosh (e+f x)}{2 (c+d x)}+\frac {\cosh (2 e+2 f x)}{8 (c+d x)}+\frac {3}{8 (c+d x)}\right )dx}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 d (c+d x)^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 a^2 \left (-\frac {3 f^2 \left (\frac {\text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{2 d}+\frac {\sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d}+\frac {\log (c+d x)}{2 d}\right )}{2 d^2}+\frac {2 f^2 \left (\frac {\text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{2 d}+\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{8 d}+\frac {\sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 d}+\frac {3 \log (c+d x)}{8 d}\right )}{d^2}-\frac {f \sinh \left (\frac {e}{2}+\frac {f x}{2}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 d (c+d x)^2}\right )\) |
Input:
Int[(a + a*Cosh[e + f*x])^2/(c + d*x)^3,x]
Output:
4*a^2*(-1/2*Cosh[e/2 + (f*x)/2]^4/(d*(c + d*x)^2) - (f*Cosh[e/2 + (f*x)/2] ^3*Sinh[e/2 + (f*x)/2])/(d^2*(c + d*x)) - (3*f^2*((Cosh[e - (c*f)/d]*CoshI ntegral[(c*f)/d + f*x])/(2*d) + Log[c + d*x]/(2*d) + (Sinh[e - (c*f)/d]*Si nhIntegral[(c*f)/d + f*x])/(2*d)))/(2*d^2) + (2*f^2*((Cosh[e - (c*f)/d]*Co shIntegral[(c*f)/d + f*x])/(2*d) + (Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2* c*f)/d + 2*f*x])/(8*d) + (3*Log[c + d*x])/(8*d) + (Sinh[e - (c*f)/d]*SinhI ntegral[(c*f)/d + f*x])/(2*d) + (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f )/d + 2*f*x])/(8*d)))/d^2)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(617\) vs. \(2(199)=398\).
Time = 2.48 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.99
method | result | size |
risch | \(\frac {f^{3} a^{2} {\mathrm e}^{-f x -e} x}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} a^{2} {\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a^{2} {\mathrm e}^{-f x -e}}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {f^{2} a^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} a^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} a^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {3 a^{2}}{4 d \left (d x +c \right )^{2}}+\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}-\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}\) | \(618\) |
Input:
int((a+a*cosh(f*x+e))^2/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/2*f^3*a^2*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^2* 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^2*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^2/d^3*exp((c*f-d*e)/d)*Ei( 1,f*x+e+(c*f-d*e)/d)-1/2*f^2*a^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*f^2*a^2/ d^3*exp(f*x+e)/(c*f/d+f*x)-1/2*f^2*a^2/d^3*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-( c*f-d*e)/d)-3/4*a^2/d/(d*x+c)^2+1/4*f^3*a^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*a^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*a^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*a^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*a^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2-1/4*f^2*a^2/d^3*exp(2*f*x+2*e) /(c*f/d+f*x)-1/2*f^2*a^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. 596 vs. \(2 (199) = 398\).
Time = 0.12 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.88 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {a^{2} d^{2} \cosh \left (f x + e\right )^{2} + a^{2} d^{2} \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d^{2} \cosh \left (f x + e\right ) + 3 \, a^{2} d^{2} - 2 \, {\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - 2 \, {\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 4 \, {\left (a^{2} d^{2} f x + a^{2} c d f + {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")
Output:
-1/4*(a^2*d^2*cosh(f*x + e)^2 + a^2*d^2*sinh(f*x + e)^2 + 4*a^2*d^2*cosh(f *x + e) + 3*a^2*d^2 - 2*((a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2) *Ei((d*f*x + c*f)/d) + (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*E i(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - 2*((a^2*d^2*f^2*x^2 + 2*a^2*c* d*f^2*x + a^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d) + (a^2*d^2*f^2*x^2 + 2*a^2*c* d*f^2*x + a^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d))*cosh(-2*(d*e - c*f)/d) + 4* (a^2*d^2*f*x + a^2*c*d*f + (a^2*d^2*f*x + a^2*c*d*f)*cosh(f*x + e))*sinh(f *x + e) + 2*((a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei((d*f*x + c*f)/d) - (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + 2*((a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^ 2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d) - (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^ 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)
\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=a^{2} \left (\int \frac {2 \cosh {\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {\cosh ^{2}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {1}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right ) \] Input:
integrate((a+a*cosh(f*x+e))**2/(d*x+c)**3,x)
Output:
a**2*(Integral(2*cosh(e + f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x **3), x) + Integral(cosh(e + f*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(1/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 ), x))
Time = 0.13 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} + \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")
Output:
-1/4*a^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^2*(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. 682 vs. \(2 (199) = 398\).
Time = 0.14 (sec) , antiderivative size = 682, normalized size of antiderivative = 3.29 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="giac")
Output:
1/8*(4*a^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 4*a^2*d^2 *f^2*x^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*a^2*d^2*f^2*x^2*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 4*a^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(-2* e + 2*c*f/d) + 8*a^2*c*d*f^2*x*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 8 *a^2*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 8*a^2*c*d*f^2*x*Ei(-(d* f*x + c*f)/d)*e^(-e + c*f/d) + 8*a^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(- 2*e + 2*c*f/d) + 4*a^2*c^2*f^2*Ei(2*(d*f*x + c*f)/d)*e^(2*e - 2*c*f/d) + 4 *a^2*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) + 4*a^2*c^2*f^2*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 4*a^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(-2*e + 2*c*f/d) - 2*a^2*d^2*f*x*e^(2*f*x + 2*e) - 4*a^2*d^2*f*x*e^(f*x + e) + 4*a ^2*d^2*f*x*e^(-f*x - e) + 2*a^2*d^2*f*x*e^(-2*f*x - 2*e) - 2*a^2*c*d*f*e^( 2*f*x + 2*e) - 4*a^2*c*d*f*e^(f*x + e) + 4*a^2*c*d*f*e^(-f*x - e) + 2*a^2* c*d*f*e^(-2*f*x - 2*e) - a^2*d^2*e^(2*f*x + 2*e) - 4*a^2*d^2*e^(f*x + e) - 4*a^2*d^2*e^(-f*x - e) - a^2*d^2*e^(-2*f*x - 2*e) - 6*a^2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + a*cosh(e + f*x))^2/(c + d*x)^3,x)
Output:
int((a + a*cosh(e + f*x))^2/(c + d*x)^3, x)
\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx=\frac {a^{2} \left (e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c^{2} d +2 e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c \,d^{2} x +e^{3 e} \left (\int \frac {e^{2 f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) d^{3} x^{2}+4 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c^{2} d +8 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c \,d^{2} x +4 e^{2 e} \left (\int \frac {e^{f x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) d^{3} x^{2}+e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) c^{2} d +2 e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) c \,d^{2} x +e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) d^{3} x^{2}-3 e^{e}+4 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) c^{2} d +8 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) c \,d^{2} x +4 \left (\int \frac {1}{e^{f x} c^{3}+3 e^{f x} c^{2} d x +3 e^{f x} c \,d^{2} x^{2}+e^{f x} d^{3} x^{3}}d x \right ) d^{3} x^{2}\right )}{4 e^{e} d \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((a+a*cosh(f*x+e))^2/(d*x+c)^3,x)
Output:
(a**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)*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)*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)*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)*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)*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)*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)*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)*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)*d**3*x**2 - 3*e**e + 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)*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)*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)*d**3*x**2))/(4*e**e*d*(c**2 + 2*c*d*x + d**2*x**2))