Optimal. Leaf size=139 \[ -\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{24 x}+\frac{17}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{17}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{7 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{12 x^2}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3} \]
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Rubi [A] time = 0.0614669, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6126, 99, 151, 12, 93, 212, 206, 203} \[ -\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{24 x}+\frac{17}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{17}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{7 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{12 x^2}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{-\frac{3}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-a x)^{3/4}}{x^4 (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{1}{3} \int \frac{-\frac{7 a}{2}+2 a^2 x}{x^3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{1}{6} \int \frac{-\frac{23 a^2}{4}+\frac{7 a^3 x}{2}}{x^2 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac{1}{6} \int -\frac{51 a^3}{8 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}-\frac{1}{16} \left (17 a^3\right ) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}-\frac{1}{4} \left (17 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac{1}{8} \left (17 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac{1}{8} \left (17 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac{7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac{23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac{17}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac{17}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0218255, size = 78, normalized size = 0.56 \[ \frac{(1-a x)^{3/4} \left (34 a^3 x^3 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )-23 a^3 x^3-9 a^2 x^2+6 a x-8\right )}{24 x^3 (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.107, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77509, size = 355, normalized size = 2.55 \begin{align*} \frac{102 \, a^{3} x^{3} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \,{\left (23 \, a^{3} x^{3} - 37 \, a^{2} x^{2} + 22 \, a x - 8\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}} \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{1}{x^{4} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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