Integrand size = 19, antiderivative size = 141 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {6 b^2 \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {b^2 x^3 \sinh (c+d x)}{d}+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \]
1/2*a^2*d^2*Chi(d*x)*cosh(c)-6*b^2*cosh(d*x+c)/d^4-1/2*a^2*cosh(d*x+c)/x^2 -3*b^2*x^2*cosh(d*x+c)/d^2+1/2*a^2*d^2*Shi(d*x)*sinh(c)+2*a*b*sinh(d*x+c)/ d-1/2*a^2*d*sinh(d*x+c)/x+6*b^2*x*sinh(d*x+c)/d^3+b^2*x^3*sinh(d*x+c)/d
Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {12 b^2 \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{x^2}-\frac {6 b^2 x^2 \cosh (c+d x)}{d^2}+a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{x}+\frac {12 b^2 x \sinh (c+d x)}{d^3}+\frac {2 b^2 x^3 \sinh (c+d x)}{d}+a^2 d^2 \sinh (c) \text {Shi}(d x)\right ) \]
((-12*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/x^2 - (6*b^2*x^2*Cosh[c + d*x])/d^2 + a^2*d^2*Cosh[c]*CoshIntegral[d*x] + (4*a*b*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/x + (12*b^2*x*Sinh[c + d*x])/d^3 + (2*b^2*x^3*Sin h[c + d*x])/d + a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2
Time = 0.46 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5810, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5810 |
\(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{x^3}+2 a b \cosh (c+d x)+b^2 x^3 \cosh (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {2 a b \sinh (c+d x)}{d}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {b^2 x^3 \sinh (c+d x)}{d}\) |
(-6*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/(2*x^2) - (3*b^2*x^2*Cosh [c + d*x])/d^2 + (a^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (2*a*b*Sinh[c + d *x])/d - (a^2*d*Sinh[c + d*x])/(2*x) + (6*b^2*x*Sinh[c + d*x])/d^3 + (b^2* x^3*Sinh[c + d*x])/d + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2
3.1.91.3.1 Defintions of rubi rules used
Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p _.), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(133)=266\).
Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.99
method | result | size |
risch | \(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x^{2}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x^{2}-2 \,{\mathrm e}^{d x +c} b^{2} d^{3} x^{5}+2 \,{\mathrm e}^{-d x -c} b^{2} d^{3} x^{5}+{\mathrm e}^{d x +c} a^{2} d^{5} x +6 \,{\mathrm e}^{d x +c} b^{2} d^{2} x^{4}-{\mathrm e}^{-d x -c} a^{2} d^{5} x +6 \,{\mathrm e}^{-d x -c} b^{2} d^{2} x^{4}-4 \,{\mathrm e}^{d x +c} a b \,d^{3} x^{2}+4 \,{\mathrm e}^{-d x -c} a b \,d^{3} x^{2}+d^{4} {\mathrm e}^{d x +c} a^{2}-12 \,{\mathrm e}^{d x +c} b^{2} d \,x^{3}+d^{4} {\mathrm e}^{-d x -c} a^{2}+12 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{3}+12 \,{\mathrm e}^{d x +c} b^{2} x^{2}+12 \,{\mathrm e}^{-d x -c} b^{2} x^{2}}{4 d^{4} x^{2}}\) | \(281\) |
meijerg | \(\frac {8 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {8 i b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {2 a b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {2 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}-\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(329\) |
-1/4/d^4*(exp(c)*Ei(1,-d*x)*a^2*d^6*x^2+exp(-c)*Ei(1,d*x)*a^2*d^6*x^2-2*ex p(d*x+c)*b^2*d^3*x^5+2*exp(-d*x-c)*b^2*d^3*x^5+exp(d*x+c)*a^2*d^5*x+6*exp( d*x+c)*b^2*d^2*x^4-exp(-d*x-c)*a^2*d^5*x+6*exp(-d*x-c)*b^2*d^2*x^4-4*exp(d *x+c)*a*b*d^3*x^2+4*exp(-d*x-c)*a*b*d^3*x^2+d^4*exp(d*x+c)*a^2-12*exp(d*x+ c)*b^2*d*x^3+d^4*exp(-d*x-c)*a^2+12*exp(-d*x-c)*b^2*d*x^3+12*exp(d*x+c)*b^ 2*x^2+12*exp(-d*x-c)*b^2*x^2)/x^2
Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, {\left (6 \, b^{2} d^{2} x^{4} + a^{2} d^{4} + 12 \, b^{2} x^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) + a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (2 \, b^{2} d^{3} x^{5} - a^{2} d^{5} x + 4 \, a b d^{3} x^{2} + 12 \, b^{2} d x^{3}\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) - a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d^{4} x^{2}} \]
-1/4*(2*(6*b^2*d^2*x^4 + a^2*d^4 + 12*b^2*x^2)*cosh(d*x + c) - (a^2*d^6*x^ 2*Ei(d*x) + a^2*d^6*x^2*Ei(-d*x))*cosh(c) - 2*(2*b^2*d^3*x^5 - a^2*d^5*x + 4*a*b*d^3*x^2 + 12*b^2*d*x^3)*sinh(d*x + c) - (a^2*d^6*x^2*Ei(d*x) - a^2* d^6*x^2*Ei(-d*x))*sinh(c))/(d^4*x^2)
\[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]
Time = 0.24 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{8} \, {\left (2 \, a^{2} d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 2 \, a^{2} d e^{c} \Gamma \left (-1, -d x\right ) - \frac {8 \, {\left (d x e^{c} - e^{c}\right )} a b e^{\left (d x\right )}}{d^{2}} - \frac {8 \, {\left (d x + 1\right )} a b e^{\left (-d x - c\right )}}{d^{2}} - \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{5}} - \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{d^{5}}\right )} d + \frac {1}{4} \, {\left (b^{2} x^{4} + 8 \, a b x - \frac {2 \, a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]
1/8*(2*a^2*d*e^(-c)*gamma(-1, d*x) + 2*a^2*d*e^c*gamma(-1, -d*x) - 8*(d*x* e^c - e^c)*a*b*e^(d*x)/d^2 - 8*(d*x + 1)*a*b*e^(-d*x - c)/d^2 - (d^4*x^4*e ^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*b^2*e^(d*x)/d^5 - (d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*b^2*e^(-d*x - c)/d^5)* d + 1/4*(b^2*x^4 + 8*a*b*x - 2*a^2/x^2)*cosh(d*x + c)
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (133) = 266\).
Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} d^{3} x^{5} e^{\left (d x + c\right )} - 2 \, b^{2} d^{3} x^{5} e^{\left (-d x - c\right )} - a^{2} d^{5} x e^{\left (d x + c\right )} - 6 \, b^{2} d^{2} x^{4} e^{\left (d x + c\right )} + a^{2} d^{5} x e^{\left (-d x - c\right )} - 6 \, b^{2} d^{2} x^{4} e^{\left (-d x - c\right )} + 4 \, a b d^{3} x^{2} e^{\left (d x + c\right )} - 4 \, a b d^{3} x^{2} e^{\left (-d x - c\right )} - a^{2} d^{4} e^{\left (d x + c\right )} + 12 \, b^{2} d x^{3} e^{\left (d x + c\right )} - a^{2} d^{4} e^{\left (-d x - c\right )} - 12 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - 12 \, b^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b^{2} x^{2} e^{\left (-d x - c\right )}}{4 \, d^{4} x^{2}} \]
1/4*(a^2*d^6*x^2*Ei(-d*x)*e^(-c) + a^2*d^6*x^2*Ei(d*x)*e^c + 2*b^2*d^3*x^5 *e^(d*x + c) - 2*b^2*d^3*x^5*e^(-d*x - c) - a^2*d^5*x*e^(d*x + c) - 6*b^2* d^2*x^4*e^(d*x + c) + a^2*d^5*x*e^(-d*x - c) - 6*b^2*d^2*x^4*e^(-d*x - c) + 4*a*b*d^3*x^2*e^(d*x + c) - 4*a*b*d^3*x^2*e^(-d*x - c) - a^2*d^4*e^(d*x + c) + 12*b^2*d*x^3*e^(d*x + c) - a^2*d^4*e^(-d*x - c) - 12*b^2*d*x^3*e^(- d*x - c) - 12*b^2*x^2*e^(d*x + c) - 12*b^2*x^2*e^(-d*x - c))/(d^4*x^2)
Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^3} \,d x \]