Optimal. Leaf size=232 \[ \cosh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]
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Rubi [A] time = 0.517844, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5365, 5293, 3303, 3298, 3301} \[ \cosh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 5365
Rule 5293
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\cosh \left (a+b \sqrt [3]{x}\right )}{-c+x} \, dx,x,c+d x\right )\\ &=3 \operatorname{Subst}\left (\int \frac{x^2 \cosh (a+b x)}{-c+x^3} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{\cosh (a+b x)}{3 \left (\sqrt [3]{c}-x\right )}-\frac{\cosh (a+b x)}{3 \left (-\sqrt [3]{-1} \sqrt [3]{c}-x\right )}-\frac{\cosh (a+b x)}{3 \left ((-1)^{2/3} \sqrt [3]{c}-x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\left (\cosh \left (a+b \sqrt [3]{c}\right ) \operatorname{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 )\right )-\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname{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 )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname{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 )+\sinh \left (a+b \sqrt [3]{c}\right ) \operatorname{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 \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \operatorname{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 \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \operatorname{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 )\\ &=\cosh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Chi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )+\cosh \left (a+(-1)^{2/3} 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+b \sqrt [3]{c}\right ) \text{Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sinh \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 )+\sinh \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*}
Mathematica [C] time = 0.0643569, size = 231, normalized size = 1. \[ \frac{1}{2} \left (\text{RootSum}\left [c-\text{$\#$1}^3\& ,-\sinh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\sinh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-\cosh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]+\text{RootSum}\left [c-\text{$\#$1}^3\& ,\sinh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\sinh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\cosh \left ( a+b\sqrt [3]{dx+c} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03753, size = 1538, normalized size = 6.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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