Optimal. Leaf size=776 \[ \text{result too large to display} \]
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Rubi [A] time = 2.65879, antiderivative size = 776, normalized size of antiderivative = 1., number of steps used = 71, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {5291, 5279, 5293, 3297, 3303, 3298, 3301, 5281, 5292, 5290} \[ -\frac{\sqrt [3]{-1} \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 )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{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 )}{27 a^{5/3} b^{4/3}}+\frac{\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 )}{27 a^{5/3} b^{4/3}}+\frac{\sqrt [3]{-1} \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 )}{27 a^{5/3} b^{4/3}}+\frac{\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 )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{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 )}{27 a^{5/3} b^{4/3}}-\frac{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 b^2}-\frac{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 b^2}-\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 b^2}+\frac{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 b^2}-\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 b^2}-\frac{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 b^2}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{\cosh (c+d x)}{18 a b^2 x^2}+\frac{d \sinh (c+d x)}{18 a b^2 x}-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 5291
Rule 5279
Rule 5293
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5281
Rule 5292
Rule 5290
Rubi steps
\begin{align*} \int \frac{x^3 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{\int \frac{\cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}+\frac{d \int \frac{x \sinh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{\int \frac{\cosh (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 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{\int \left (\frac{\cosh (c+d x)}{a x^3}-\frac{b \cosh (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}-\frac{b x^2 \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{\int \frac{\cosh (c+d x)}{x^3} \, dx}{9 a b^2}+\frac{\int \frac{\cosh (c+d x)}{a+b x^3} \, dx}{9 a b}+\frac{d^2 \int \frac{\cosh (c+d x)}{x} \, dx}{18 a b^2}-\frac{d^2 \int \frac{x^2 \cosh (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=\frac{\cosh (c+d x)}{18 a b^2 x^2}-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{\int \left (-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cosh (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 \int \frac{\sinh (c+d x)}{x^2} \, dx}{18 a b^2}-\frac{d^2 \int \left (\frac{\cosh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cosh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cosh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}+\frac{\left (d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{18 a b^2}+\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{18 a b^2}\\ &=\frac{\cosh (c+d x)}{18 a b^2 x^2}-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{18 a b^2}+\frac{d \sinh (c+d x)}{18 a b^2 x}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{18 a b^2}-\frac{\int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\int \frac{\cosh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\int \frac{\cosh (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)}{x} \, dx}{18 a b^2}-\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac{d^2 \int \frac{\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac{d^2 \int \frac{\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac{\cosh (c+d x)}{18 a b^2 x^2}-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{18 a b^2}+\frac{d \sinh (c+d x)}{18 a b^2 x}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{18 a b^2}-\frac{\left (d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{18 a b^2}-\frac{\cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\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 \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 b^{5/3}}-\frac{\cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 (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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac{\cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 (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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{18 a b^2}-\frac{\sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\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 \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 b^{5/3}}-\frac{\left (i \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}{27 a^{5/3} b}-\frac{\left (i 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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac{\left (i \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}{27 a^{5/3} b}-\frac{\left (i 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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac{\cosh (c+d x)}{18 a b^2 x^2}-\frac{x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sqrt [3]{-1} \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 )}{27 a^{5/3} b^{4/3}}-\frac{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 b^2}+\frac{(-1)^{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 )}{27 a^{5/3} b^{4/3}}-\frac{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 b^2}+\frac{\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 )}{27 a^{5/3} b^{4/3}}-\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 b^2}+\frac{d \sinh (c+d x)}{18 a b^2 x}-\frac{d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{\sqrt [3]{-1} \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 )}{27 a^{5/3} b^{4/3}}+\frac{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 b^2}+\frac{\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 )}{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 b^2}+\frac{(-1)^{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 )}{27 a^{5/3} b^{4/3}}-\frac{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 b^2}\\ \end{align*}
Mathematica [C] time = 0.636095, size = 429, normalized size = 0.55 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-\text{$\#$1}^2 d^2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\text{$\#$1}^2 d^2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\frac{6 b x \left (d x \left (a+b x^3\right ) \sinh (c+d x)+\left (b x^3-2 a\right ) \cosh (c+d x)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.112, size = 1456, normalized size = 1.9 \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 = 2.63863, size = 6494, normalized size = 8.37 \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^{3} \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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