Integrand size = 16, antiderivative size = 345 \[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=-\frac {\sqrt [3]{-1} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{2/3} \sqrt [3]{b}} \]
1/3*Chi(a^(1/3)*d/b^(1/3)+d*x)*cosh(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(1/3)-1 /3*(-1)^(1/3)*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c+(-1)^(1/3)*a^(1 /3)*d/b^(1/3))/a^(2/3)/b^(1/3)+1/3*(-1)^(2/3)*Chi(-(-1)^(2/3)*a^(1/3)*d/b^ (1/3)-d*x)*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(1/3)+1/3*Shi(a^ (1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(1/3)-1/3*(-1)^(1 /3)*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^( 1/3))/a^(2/3)/b^(1/3)+1/3*(-1)^(2/3)*Shi((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x) *sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 5.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.52 \[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{6 b} \]
(RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshInt egral[d*(x - #1)]*Sinh[c + d*#1] - Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)])/#1^2 & ] + RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + CoshIntegral[d*(x - #1)]*Si nh[c + d*#1] + Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + Sinh[c + d*#1]*Si nhIntegral[d*(x - #1)])/#1^2 & ])/(6*b)
Time = 0.70 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5804, 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.
\(\displaystyle \int \frac {\cosh (c+d x)}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 5804 |
\(\displaystyle \int \left (-\frac {\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cosh (c+d x)}{3 a^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a}\right )}-\frac {\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt [3]{-1} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
-1/3*((-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((- 1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b ^(1/3)) - d*x])/(3*a^(2/3)*b^(1/3)) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshI ntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(1/3)*Sinh [c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b ^(1/3) - d*x])/(3*a^(2/3)*b^(1/3)) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIn tegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Sinh[ c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^ (1/3) + d*x])/(3*a^(2/3)*b^(1/3))
3.1.98.3.1 Defintions of rubi rules used
Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> In t[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d }, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.41
method | result | size |
risch | \(-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}\) | \(143\) |
-1/6*d^2/b*sum(1/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf( _Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/6*d^2/b*sum(1/(_R1^2-2*_R1*c+ c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d ^3-b*c^3))
Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (237) = 474\).
Time = 0.27 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.95 \[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\text {Too large to display} \]
-1/12*((a*d^3/b)^(1/3)*(sqrt(-3) + 1)*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(- 3) + 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) - (-a*d^3/b)^(1/3)*( sqrt(-3) + 1)*Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(-a* d^3/b)^(1/3)*(sqrt(-3) + 1) - c) - (a*d^3/b)^(1/3)*(sqrt(-3) - 1)*Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) + (-a*d^3/b)^(1/3)*(sqrt(-3) - 1)*Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*( sqrt(-3) - 1))*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) + (a*d^3/b)^( 1/3)*(sqrt(-3) + 1)*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2* (a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) - (-a*d^3/b)^(1/3)*(sqrt(-3) + 1)*Ei(- d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(-a*d^3/b)^(1/3)*(sqrt (-3) + 1) - c) + (a*d^3/b)^(1/3)*(sqrt(-3) - 1)*Ei(d*x + 1/2*(a*d^3/b)^(1/ 3)*(sqrt(-3) - 1))*sinh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) - (-a*d^3/ b)^(1/3)*(sqrt(-3) - 1)*Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sin h(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) + 2*(-a*d^3/b)^(1/3)*Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) - 2*(a*d^3/b)^(1/3)*Ei(d*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) - 2*(-a*d^3/b)^(1/3)*Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) + 2*(a*d^3/b)^(1/3)*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)))/(a*d)
\[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{a + b x^{3}}\, dx \]
Timed out. \[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {\cosh (c+d x)}{a+b x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{b\,x^3+a} \,d x \]