Optimal. Leaf size=414 \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{4 \left (c d^3-e^3\right )}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{4 \left (c d^3+e^3\right )}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{\sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (c x^3+1\right )}{2 e \left (c d^3-e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (\sqrt [3]{c} x+1\right )}{2 \left (c d^3-e^3\right )}-\frac{\sqrt{3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )}{2 \left (c d^3+e^3\right )} \]
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Rubi [A] time = 0.772685, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6273, 12, 6725, 1871, 1861, 31, 634, 617, 204, 628, 260, 1860} \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{4 \left (c d^3-e^3\right )}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{4 \left (c d^3+e^3\right )}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{\sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (c x^3+1\right )}{2 e \left (c d^3-e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (\sqrt [3]{c} x+1\right )}{2 \left (c d^3-e^3\right )}-\frac{\sqrt{3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )}{2 \left (c d^3+e^3\right )} \]
Antiderivative was successfully verified.
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Rule 6273
Rule 12
Rule 6725
Rule 1871
Rule 1861
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rule 1860
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^3\right )}{(d+e x)^2} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b \int \frac{3 c x^2}{(d+e x) \left (1-c^2 x^6\right )} \, dx}{e}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{(3 b c) \int \frac{x^2}{(d+e x) \left (1-c^2 x^6\right )} \, dx}{e}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{(3 b c) \int \left (\frac{d^2 e^4}{\left (-c d^3+e^3\right ) \left (c d^3+e^3\right ) (d+e x)}+\frac{d e-e^2 x-c d^2 x^2}{2 \left (c d^3+e^3\right ) \left (-1+c x^3\right )}+\frac{d e-e^2 x+c d^2 x^2}{2 \left (c d^3-e^3\right ) \left (1+c x^3\right )}\right ) \, dx}{e}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac{(3 b c) \int \frac{d e-e^2 x+c d^2 x^2}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac{(3 b c) \int \frac{d e-e^2 x-c d^2 x^2}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac{(3 b c) \int \frac{d e-e^2 x}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac{\left (3 b c^2 d^2\right ) \int \frac{x^2}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac{(3 b c) \int \frac{d e-e^2 x}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}-\frac{\left (3 b c^2 d^2\right ) \int \frac{x^2}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )}+\frac{\left (b c^{2/3}\right ) \int \frac{2 \sqrt [3]{c} d e-e^2+\sqrt [3]{c} \left (-\sqrt [3]{c} d e-e^2\right ) x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c d^3-e^3\right )}+\frac{\left (b c^{2/3} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac{1}{1+\sqrt [3]{c} x} \, dx}{2 \left (c d^3-e^3\right )}+\frac{\left (b c^{2/3}\right ) \int \frac{-2 \sqrt [3]{c} d e-e^2-\sqrt [3]{c} \left (\sqrt [3]{c} d e-e^2\right ) x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c d^3+e^3\right )}-\frac{\left (b c \left (d-\frac{e}{\sqrt [3]{c}}\right )\right ) \int \frac{1}{1-\sqrt [3]{c} x} \, dx}{2 \left (c d^3+e^3\right )}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x\right )}{2 \left (c d^3-e^3\right )}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )}+\frac{\left (3 b c^{2/3}\right ) \int \frac{1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac{\left (b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac{-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3-e^3\right )}-\frac{\left (b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right )\right ) \int \frac{\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3+e^3\right )}-\frac{\left (3 b c^{2/3} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac{1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3+e^3\right )}\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x\right )}{2 \left (c d^3-e^3\right )}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3-e^3\right )}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3+e^3\right )}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )}+\frac{\left (3 b \sqrt [3]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}+\frac{\left (3 b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}\\ &=-\frac{\sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac{\sqrt{3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\frac{1+2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{2 \left (c d^3+e^3\right )}-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x\right )}{2 \left (c d^3-e^3\right )}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3-e^3\right )}-\frac{b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3+e^3\right )}-\frac{b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac{b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )}\\ \end{align*}
Mathematica [A] time = 0.558207, size = 534, normalized size = 1.29 \[ \frac{1}{4} \left (-\frac{4 a}{e (d+e x)}+\frac{2 b c d^2 e^2 \log \left (1-c^2 x^6\right )}{c^2 d^6-e^6}+\frac{b \sqrt [3]{c} \left (-c^{4/3} d^4 e+2 c^{5/3} d^5-c d^3 e^2-\sqrt [3]{c} d e^4-e^5\right ) \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{c^2 d^6 e-e^7}+\frac{b \sqrt [3]{c} \left (c^{4/3} d^4 e+2 c^{5/3} d^5-c d^3 e^2-\sqrt [3]{c} d e^4+e^5\right ) \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{e^7-c^2 d^6 e}-\frac{12 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac{2 b \sqrt [3]{c} \left (-c^{4/3} d^4 e+c^{5/3} d^5+c d^3 e^2+\sqrt [3]{c} d e^4-e^5\right ) \log \left (1-\sqrt [3]{c} x\right )}{e^7-c^2 d^6 e}-\frac{2 b \sqrt [3]{c} \left (c^{4/3} d^4 e+c^{5/3} d^5+c d^3 e^2+\sqrt [3]{c} d e^4+e^5\right ) \log \left (\sqrt [3]{c} x+1\right )}{e^7-c^2 d^6 e}+\frac{2 \sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x-1}{\sqrt{3}}\right )}{c^{2/3} d^2+\sqrt [3]{c} d e+e^2}-\frac{2 \sqrt{3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )}{c d^3+e^3}-\frac{4 b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 591, normalized size = 1.4 \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 [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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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(-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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