Integrand size = 19, antiderivative size = 150 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+2 a b \sinh (c) \text {Shi}(d x) \]
2*a*b*Chi(d*x)*cosh(c)-1/3*a^2*cosh(d*x+c)/x^3-1/6*a^2*d^2*cosh(d*x+c)/x-2 *b^2*x*cosh(d*x+c)/d^2+1/6*a^2*d^3*cosh(c)*Shi(d*x)+1/6*a^2*d^3*Chi(d*x)*s inh(c)+2*a*b*Shi(d*x)*sinh(c)+2*b^2*sinh(d*x+c)/d^3-1/6*a^2*d*sinh(d*x+c)/ x^2+b^2*x^2*sinh(d*x+c)/d
Time = 0.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a^2 \cosh (c+d x)}{x^3}-\frac {a^2 d^2 \cosh (c+d x)}{x}-\frac {12 b^2 x \cosh (c+d x)}{d^2}+a \text {Chi}(d x) \left (12 b \cosh (c)+a d^3 \sinh (c)\right )+\frac {12 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{x^2}+\frac {6 b^2 x^2 \sinh (c+d x)}{d}+a \left (a d^3 \cosh (c)+12 b \sinh (c)\right ) \text {Shi}(d x)\right ) \]
((-2*a^2*Cosh[c + d*x])/x^3 - (a^2*d^2*Cosh[c + d*x])/x - (12*b^2*x*Cosh[c + d*x])/d^2 + a*CoshIntegral[d*x]*(12*b*Cosh[c] + a*d^3*Sinh[c]) + (12*b^ 2*Sinh[c + d*x])/d^3 - (a^2*d*Sinh[c + d*x])/x^2 + (6*b^2*x^2*Sinh[c + d*x ])/d + a*(a*d^3*Cosh[c] + 12*b*Sinh[c])*SinhIntegral[d*x])/6
Time = 0.49 (sec) , antiderivative size = 150, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{x^4}+\frac {2 a b \cosh (c+d x)}{x}+b^2 x^2 \cosh (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}\) |
-1/3*(a^2*Cosh[c + d*x])/x^3 - (a^2*d^2*Cosh[c + d*x])/(6*x) - (2*b^2*x*Co sh[c + d*x])/d^2 + 2*a*b*Cosh[c]*CoshIntegral[d*x] + (a^2*d^3*CoshIntegral [d*x]*Sinh[c])/6 + (2*b^2*Sinh[c + d*x])/d^3 - (a^2*d*Sinh[c + d*x])/(6*x^ 2) + (b^2*x^2*Sinh[c + d*x])/d + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6 + 2 *a*b*Sinh[c]*SinhIntegral[d*x]
3.1.92.3.1 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. \(283\) vs. \(2(140)=280\).
Time = 0.31 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(-\frac {-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x^{3}+{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x^{3}+{\mathrm e}^{d x +c} a^{2} d^{5} x^{2}-6 \,{\mathrm e}^{d x +c} b^{2} d^{2} x^{5}+{\mathrm e}^{-d x -c} a^{2} d^{5} x^{2}+6 \,{\mathrm e}^{-d x -c} b^{2} d^{2} x^{5}+12 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{3} x^{3}+12 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{3} x^{3}+d^{4} {\mathrm e}^{d x +c} a^{2} x +12 \,{\mathrm e}^{d x +c} b^{2} d \,x^{4}-d^{4} {\mathrm e}^{-d x -c} a^{2} x +12 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{4}+2 \,{\mathrm e}^{d x +c} a^{2} d^{3}-12 \,{\mathrm e}^{d x +c} b^{2} x^{3}+2 \,{\mathrm e}^{-d x -c} a^{2} d^{3}+12 \,{\mathrm e}^{-d x -c} b^{2} x^{3}}{12 d^{3} x^{3}}\) | \(284\) |
| meijerg | \(\frac {4 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )+2 a b \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )-\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(347\) |
-1/12/d^3*(-exp(-c)*Ei(1,d*x)*a^2*d^6*x^3+exp(c)*Ei(1,-d*x)*a^2*d^6*x^3+ex p(d*x+c)*a^2*d^5*x^2-6*exp(d*x+c)*b^2*d^2*x^5+exp(-d*x-c)*a^2*d^5*x^2+6*ex p(-d*x-c)*b^2*d^2*x^5+12*exp(-c)*Ei(1,d*x)*a*b*d^3*x^3+12*exp(c)*Ei(1,-d*x )*a*b*d^3*x^3+d^4*exp(d*x+c)*a^2*x+12*exp(d*x+c)*b^2*d*x^4-d^4*exp(-d*x-c) *a^2*x+12*exp(-d*x-c)*b^2*d*x^4+2*exp(d*x+c)*a^2*d^3-12*exp(d*x+c)*b^2*x^3 +2*exp(-d*x-c)*a^2*d^3+12*exp(-d*x-c)*b^2*x^3)/x^3
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (a^{2} d^{5} x^{2} + 12 \, b^{2} d x^{4} + 2 \, a^{2} d^{3}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (6 \, b^{2} d^{2} x^{5} - a^{2} d^{4} x + 12 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, d^{3} x^{3}} \]
-1/12*(2*(a^2*d^5*x^2 + 12*b^2*d*x^4 + 2*a^2*d^3)*cosh(d*x + c) - ((a^2*d^ 6 + 12*a*b*d^3)*x^3*Ei(d*x) - (a^2*d^6 - 12*a*b*d^3)*x^3*Ei(-d*x))*cosh(c) - 2*(6*b^2*d^2*x^5 - a^2*d^4*x + 12*b^2*x^3)*sinh(d*x + c) - ((a^2*d^6 + 12*a*b*d^3)*x^3*Ei(d*x) + (a^2*d^6 - 12*a*b*d^3)*x^3*Ei(-d*x))*sinh(c))/(d ^3*x^3)
\[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \, {\left ({\left (d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} a^{2} - b^{2} {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} - \frac {4 \, a b \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} + \frac {6 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + \frac {1}{3} \, {\left (b^{2} x^{3} + 2 \, a b \log \left (x^{3}\right ) - \frac {a^{2}}{x^{3}}\right )} \cosh \left (d x + c\right ) \]
1/6*((d^2*e^(-c)*gamma(-2, d*x) - d^2*e^c*gamma(-2, -d*x))*a^2 - b^2*((d^3 *x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d ^2*x^2 + 6*d*x + 6)*e^(-d*x - c)/d^4) - 4*a*b*cosh(d*x + c)*log(x^3)/d + 6 *(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*a*b/d)*d + 1/3*(b^2*x^3 + 2*a*b*log(x^3) - a^2/x^3)*cosh(d*x + c)
Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^{2} d^{6} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{6} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{5} x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} d^{2} x^{5} e^{\left (d x + c\right )} + a^{2} d^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} d^{2} x^{5} e^{\left (-d x - c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{4} x e^{\left (d x + c\right )} + 12 \, b^{2} d x^{4} e^{\left (d x + c\right )} - a^{2} d^{4} x e^{\left (-d x - c\right )} + 12 \, b^{2} d x^{4} e^{\left (-d x - c\right )} + 2 \, a^{2} d^{3} e^{\left (d x + c\right )} - 12 \, b^{2} x^{3} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} e^{\left (-d x - c\right )} + 12 \, b^{2} x^{3} e^{\left (-d x - c\right )}}{12 \, d^{3} x^{3}} \]
-1/12*(a^2*d^6*x^3*Ei(-d*x)*e^(-c) - a^2*d^6*x^3*Ei(d*x)*e^c + a^2*d^5*x^2 *e^(d*x + c) - 6*b^2*d^2*x^5*e^(d*x + c) + a^2*d^5*x^2*e^(-d*x - c) + 6*b^ 2*d^2*x^5*e^(-d*x - c) - 12*a*b*d^3*x^3*Ei(-d*x)*e^(-c) - 12*a*b*d^3*x^3*E i(d*x)*e^c + a^2*d^4*x*e^(d*x + c) + 12*b^2*d*x^4*e^(d*x + c) - a^2*d^4*x* e^(-d*x - c) + 12*b^2*d*x^4*e^(-d*x - c) + 2*a^2*d^3*e^(d*x + c) - 12*b^2* x^3*e^(d*x + c) + 2*a^2*d^3*e^(-d*x - c) + 12*b^2*x^3*e^(-d*x - c))/(d^3*x ^3)
Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^4} \,d x \]