Optimal. Leaf size=373 \[ \frac{(-1)^{2/3} a^{2/3} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}+\frac{a^{2/3} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{(-1)^{2/3} a^{2/3} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}+\frac{a^{2/3} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.937188, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5293, 3296, 2638, 3303, 3298, 3301} \[ \frac{(-1)^{2/3} a^{2/3} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}+\frac{a^{2/3} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{(-1)^{2/3} a^{2/3} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}+\frac{a^{2/3} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 5293
Rule 3296
Rule 2638
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^4 \cosh (c+d x)}{a+b x^3} \, dx &=\int \left (\frac{x \cosh (c+d x)}{b}-\frac{a x \cosh (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int x \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{x \cosh (c+d x)}{a+b x^3} \, dx}{b}\\ &=\frac{x \sinh (c+d x)}{b d}-\frac{a \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}{b}-\frac{\int \sinh (c+d x) \, dx}{b d}\\ &=-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d}+\frac{a^{2/3} \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{4/3}}-\frac{\left (\sqrt [3]{-1} a^{2/3}\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b^{4/3}}+\frac{\left ((-1)^{2/3} a^{2/3}\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b^{4/3}}\\ &=-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d}+\frac{\left (a^{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 b^{4/3}}-\frac{\left (\sqrt [3]{-1} a^{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 b^{4/3}}+\frac{\left ((-1)^{2/3} a^{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 b^{4/3}}+\frac{\left (a^{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 b^{4/3}}-\frac{\left ((-1)^{5/6} a^{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 b^{4/3}}-\frac{\left (\sqrt [6]{-1} a^{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 b^{4/3}}\\ &=-\frac{\cosh (c+d x)}{b d^2}+\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{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 b^{5/3}}+\frac{a^{2/3} \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^{5/3}}+\frac{x \sinh (c+d x)}{b d}-\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}+\frac{a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{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 b^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.462271, size = 213, normalized size = 0.57 \[ -\frac{a d^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+a d^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+6 b (\cosh (c+d x)-d x \sinh (c+d x))}{6 b^2 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.087, size = 925, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13585, size = 2627, normalized size = 7.04 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{x^{4} \cosh \left (d x + c\right )}{b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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