Optimal. Leaf size=781 \[ \text{result too large to display} \]
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Rubi [A] time = 1.4493, antiderivative size = 781, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {5289, 5278, 5292, 3297, 3303, 3298, 3301, 5280, 5293} \[ \frac{(-1)^{2/3} d^2 \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 )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d \sinh \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 )}{27 a^{5/3} b^{4/3}}-\frac{\sqrt [3]{-1} d \sinh \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 )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \sinh \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 )}{27 a^{5/3} b^{4/3}}-\frac{(-1)^{2/3} d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d^2 \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 )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{\sqrt [3]{-1} d \cosh \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 )}{27 a^{5/3} b^{4/3}}+\frac{d \cosh \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 )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \cosh \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 )}{27 a^{5/3} b^{4/3}}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d \sinh (c+d x)}{18 a b^2 x^2}-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 5289
Rule 5278
Rule 5292
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5280
Rule 5293
Rubi steps
\begin{align*} \int \frac{x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d \int \frac{\sinh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \int \frac{\sinh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{9 b^2}+\frac{d^2 \int \frac{\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \int \left (\frac{\sinh (c+d x)}{a x^3}-\frac{b \sinh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{9 b^2}+\frac{d^2 \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}{18 b^2}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \int \frac{\sinh (c+d x)}{x^3} \, dx}{9 a b^2}+\frac{d \int \frac{\sinh (c+d x)}{a+b x^3} \, dx}{9 a b}+\frac{d^2 \int \frac{\cosh (c+d x)}{x^2} \, dx}{18 a b^2}-\frac{d^2 \int \frac{x \cosh (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=-\frac{d^2 \cosh (c+d x)}{18 a b^2 x}-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d \sinh (c+d x)}{18 a b^2 x^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d \int \left (-\frac{\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a b}-\frac{d^2 \int \frac{\cosh (c+d x)}{x^2} \, dx}{18 a b^2}-\frac{d^2 \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}{18 a b}+\frac{d^3 \int \frac{\sinh (c+d x)}{x} \, dx}{18 a b^2}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d \sinh (c+d x)}{18 a b^2 x^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \int \frac{\sinh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{d \int \frac{\sinh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{d \int \frac{\sinh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}+\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left (\sqrt [3]{-1} d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}+\frac{\left ((-1)^{2/3} d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{d^3 \int \frac{\sinh (c+d x)}{x} \, dx}{18 a b^2}+\frac{\left (d^3 \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{18 a b^2}+\frac{\left (d^3 \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{18 a b^2}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d^3 \text{Chi}(d x) \sinh (c)}{18 a b^2}+\frac{d \sinh (c+d x)}{18 a b^2 x^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d^3 \cosh (c) \text{Shi}(d x)}{18 a b^2}-\frac{\left (d^3 \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{18 a b^2}-\frac{\left (d \cosh \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}{27 a^{5/3} b}+\frac{\left (d^2 \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}{54 a^{4/3} b^{4/3}}-\frac{\left (i d \cosh \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}{27 a^{5/3} b}-\frac{\left (\sqrt [3]{-1} d^2 \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}{54 a^{4/3} b^{4/3}}-\frac{\left (i d \cosh \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}{27 a^{5/3} b}+\frac{\left ((-1)^{2/3} d^2 \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}{54 a^{4/3} b^{4/3}}-\frac{\left (d^3 \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{18 a b^2}-\frac{\left (d \sinh \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}{27 a^{5/3} b}+\frac{\left (d^2 \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}{54 a^{4/3} b^{4/3}}-\frac{\left (d \sinh \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}{27 a^{5/3} b}-\frac{\left ((-1)^{5/6} d^2 \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}{54 a^{4/3} b^{4/3}}-\frac{\left (d \sinh \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}{27 a^{5/3} b}-\frac{\left (\sqrt [6]{-1} d^2 \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}{54 a^{4/3} b^{4/3}}\\ &=-\frac{\cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{(-1)^{2/3} d^2 \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 )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}-\frac{\sqrt [3]{-1} d \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac{d \sinh (c+d x)}{18 a b^2 x^2}-\frac{d \sinh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{\sqrt [3]{-1} d \cosh \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 )}{27 a^{5/3} b^{4/3}}-\frac{(-1)^{2/3} d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{d \cosh \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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a^{4/3} b^{5/3}}+\frac{(-1)^{2/3} d \cosh \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 )}{27 a^{5/3} b^{4/3}}-\frac{\sqrt [3]{-1} d^2 \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 )}{54 a^{4/3} b^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.421524, size = 423, normalized size = 0.54 \[ -\frac{d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-\text{$\#$1} d \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1} d \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\text{$\#$1} d \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\text{$\#$1} d \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1} d \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1} d \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\text{$\#$1} d \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\text{$\#$1} d \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\frac{6 b \cosh (d x) \left (d x \sinh (c) \left (a+b x^3\right )-3 a \cosh (c)\right )}{\left (a+b x^3\right )^2}-\frac{6 b \sinh (d x) \left (d x \cosh (c) \left (a+b x^3\right )-3 a \sinh (c)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.086, size = 994, normalized size = 1.3 \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 = 2.53715, size = 6869, normalized size = 8.8 \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^{2} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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