Integrand size = 19, antiderivative size = 381 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=-\frac {\cosh (c+d x)}{a x}+\frac {(-1)^{2/3} \sqrt [3]{b} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {(-1)^{2/3} \sqrt [3]{b} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{4/3}} \]
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Time = 0.44 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5401, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {(-1)^{2/3} \sqrt [3]{b} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{4/3}}+\frac {d \sinh (c) \text {Chi}(d x)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\cosh (c+d x)}{a x} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a x^2}-\frac {b x \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a}-\frac {b \int \frac {x \cosh (c+d x)}{a+b x^3} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}-\frac {b \int \left (-\frac {\cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}+\frac {b^{2/3} \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3}\right ) \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {\left ((-1)^{2/3} b^{2/3}\right ) \int \frac {\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}+\frac {\left (b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\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 a^{4/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {\left ((-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {\left (b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\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 a^{4/3}}-\frac {\left ((-1)^{5/6} b^{2/3} \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]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {\left (\sqrt [6]{-1} b^{2/3} \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 )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}} \\ & = -\frac {\cosh (c+d x)}{a x}+\frac {(-1)^{2/3} \sqrt [3]{b} \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^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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^{4/3}}+\frac {\sqrt [3]{b} \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^{4/3}}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {(-1)^{2/3} \sqrt [3]{b} \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^{4/3}}+\frac {\sqrt [3]{b} \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^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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^{4/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.56 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=-\frac {6 \cosh (c+d x)+x \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]+x \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]-6 d x \text {Chi}(d x) \sinh (c)-6 d x \cosh (c) \text {Shi}(d x)}{6 a x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {3 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) x d -3 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) x d -\left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R2}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R2} -c \right )}{\textit {\_R2} -c}\right ) x d -\left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R2}} \operatorname {Ei}_{1}\left (d x -\textit {\_R2} +c \right )}{\textit {\_R2} -c}\right ) x d +3 \,{\mathrm e}^{-d x -c}+3 \,{\mathrm e}^{d x +c}}{6 a x}\) | \(175\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1154 vs. \(2 (273) = 546\).
Time = 0.28 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.03 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{2} \left (a + b x^{3}\right )}\, dx \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,\left (b\,x^3+a\right )} \,d x \]
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