Optimal. Leaf size=695 \[ -\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a^{4/3} b^{2/3}}+\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a b}-\frac {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 )}{9 a b}-\frac {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 )}{9 a b}+\frac {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 )}{9 a b}-\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 )}{9 a b}-\frac {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 )}{9 a b}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\cosh (c+d x)}{3 a b x} \]
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Rubi [A] time = 1.32, antiderivative size = 695, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5291, 5293, 3297, 3303, 3298, 3301, 5292} \[ -\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a^{4/3} b^{2/3}}+\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a b}-\frac {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 )}{9 a b}-\frac {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 )}{9 a b}+\frac {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 )}{9 a b}-\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 )}{9 a b}-\frac {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 )}{9 a b}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\cosh (c+d x)}{3 a b x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5291
Rule 5292
Rule 5293
Rubi steps
\begin {align*} \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx &=-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {\int \frac {\cosh (c+d x)}{x^2 \left (a+b x^3\right )} \, dx}{3 b}+\frac {d \int \frac {\sinh (c+d x)}{x \left (a+b x^3\right )} \, dx}{3 b}\\ &=-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {\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}{3 b}+\frac {d \int \left (\frac {\sinh (c+d x)}{a x}-\frac {b x^2 \sinh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\int \frac {x \cosh (c+d x)}{a+b x^3} \, dx}{3 a}-\frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{3 a b}-\frac {d \int \frac {x^2 \sinh (c+d x)}{a+b x^3} \, dx}{3 a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{3 a b}\\ &=\frac {\cosh (c+d x)}{3 a b x}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\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}{3 a}-\frac {d \int \left (\frac {\sinh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sinh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sinh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{3 a}-\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{3 a b}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{3 a b}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{3 a b}\\ &=\frac {\cosh (c+d x)}{3 a b x}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {d \text {Chi}(d x) \sinh (c)}{3 a b}+\frac {d \cosh (c) \text {Shi}(d x)}{3 a b}-\frac {\int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}-\frac {(-1)^{2/3} \int \frac {\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {d \int \frac {\sinh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{3 a b}-\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{3 a b}\\ &=\frac {\cosh (c+d x)}{3 a b x}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}-\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}{9 a^{4/3} \sqrt [3]{b}}-\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}{9 a b^{2/3}}+\frac {\left (\sqrt [3]{-1} \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}{9 a^{4/3} \sqrt [3]{b}}-\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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {\left ((-1)^{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}{9 a^{4/3} \sqrt [3]{b}}-\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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\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}{9 a^{4/3} \sqrt [3]{b}}-\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}{9 a b^{2/3}}+\frac {\left ((-1)^{5/6} \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}{9 a^{4/3} \sqrt [3]{b}}-\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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}+\frac {\left (\sqrt [6]{-1} \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}{9 a^{4/3} \sqrt [3]{b}}-\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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}\\ &=\frac {\cosh (c+d x)}{3 a b x}-\frac {\cosh (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a b}-\frac {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 )}{9 a b}-\frac {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 )}{9 a b}+\frac {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 )}{9 a b}+\frac {(-1)^{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 )}{9 a^{4/3} b^{2/3}}-\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 )}{9 a b}-\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 )}{9 a^{4/3} b^{2/3}}-\frac {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 )}{9 a b}+\frac {\sqrt [3]{-1} \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 )}{9 a^{4/3} b^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 387, normalized size = 0.56 \[ \frac {\left (a+b x^3\right ) \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-\text {$\#$1} d \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1} d \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\text {$\#$1} d \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\text {$\#$1} d \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]-\left (a+b x^3\right ) \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1} d \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1} d \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\text {$\#$1} d \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\text {$\#$1} d \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]+6 b x^2 \cosh (c+d x)}{18 a b \left (a+b x^3\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.94, size = 2135, normalized size = 3.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 395, normalized size = 0.57 \[ \frac {d^{3} {\mathrm e}^{-d x -c} x^{2}}{6 a \left (b \,d^{3} x^{3}+a \,d^{3}\right )}-\frac {d \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-\textit {\_R1} c +\textit {\_R1} +c \right ) {\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}+\frac {d c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c +2\right ) {\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}+\frac {d^{3} {\mathrm e}^{d x +c} x^{2}}{6 a \left (b \,d^{3} x^{3}+a \,d^{3}\right )}+\frac {d \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-\textit {\_R1} c -\textit {\_R1} -c \right ) {\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}-\frac {d c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c -2\right ) {\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x e^{\left (d x + 2 \, c\right )} - x e^{\left (-d x\right )}}{2 \, {\left (b^{2} d x^{6} e^{c} + 2 \, a b d x^{3} e^{c} + a^{2} d e^{c}\right )}} + \frac {1}{2} \, \int \frac {{\left (5 \, b x^{3} e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{b^{3} d x^{9} + 3 \, a b^{2} d x^{6} + 3 \, a^{2} b d x^{3} + a^{3} d}\,{d x} - \frac {1}{2} \, \int \frac {{\left (5 \, b x^{3} - a\right )} e^{\left (-d x\right )}}{b^{3} d x^{9} e^{c} + 3 \, a b^{2} d x^{6} e^{c} + 3 \, a^{2} b d x^{3} e^{c} + a^{3} d e^{c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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