Integrand size = 18, antiderivative size = 232 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+b \sqrt [3]{c}\right )+\text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )+\text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5472, 5400, 3384, 3379, 3382} \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\sinh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )+\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5400
Rule 5472
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sinh \left (a+b \sqrt [3]{x}\right )}{-c+x} \, dx,x,c+d x\right ) \\ & = 3 \text {Subst}\left (\int \frac {x^2 \sinh (a+b x)}{-c+x^3} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = 3 \text {Subst}\left (\int \left (-\frac {\sinh (a+b x)}{3 \left (\sqrt [3]{c}-x\right )}-\frac {\sinh (a+b x)}{3 \left (-\sqrt [3]{-1} \sqrt [3]{c}-x\right )}-\frac {\sinh (a+b x)}{3 \left ((-1)^{2/3} \sqrt [3]{c}-x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = -\text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\text {Subst}\left (\int \frac {\sinh (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\text {Subst}\left (\int \frac {\sinh (a+b x)}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = \cosh \left (a+b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\left (i \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\left (i \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cos \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cos \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right ) \sinh \left (a+b \sqrt [3]{c}\right )+\text {Chi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )+\text {Chi}\left (-(-1)^{2/3} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \left (-\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})-\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]+\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})+\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right ) \]
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\[\int \frac {\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (182) = 364\).
Time = 0.26 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.17 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=-\frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \]
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\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \]
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\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \]
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\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \]
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