Integrand size = 19, antiderivative size = 283 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\frac {\cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b}-\frac {\sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b}+\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b} \]
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Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5401, 3384, 3379, 3382} \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}-\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}} \\ & = \frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cosh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cos \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cos \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sinh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\left (i \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\left (i \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}} \\ & = \frac {\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 b}+\frac {\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 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 b}-\frac {\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 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 b}+\frac {\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 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 5.04 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.60 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]}{6 b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.49
method | result | size |
risch | \(-\frac {c^{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 {c^{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 {c \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 {\textit {\_R1} \,{\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}+\frac {c \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 {\textit {\_R1} \,{\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}-\frac {\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 {\textit {\_R1}^{2} {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 b}-\frac {\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 {\textit {\_R1}^{2} {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 b}\) | \(423\) |
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Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (207) = 414\).
Time = 0.27 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.77 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\frac {{\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) + {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) + {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) + {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) + {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) + {\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) - {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) - {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) - {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) - {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right )}{6 \, b} \]
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\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\int \frac {x^{2} \cosh {\left (c + d x \right )}}{a + b x^{3}}\, dx \]
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\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{b x^{3} + a} \,d x } \]
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\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{b x^{3} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^3+a} \,d x \]
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