Optimal. Leaf size=248 \[ \frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\sqrt {2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )+\sqrt {2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 248, normalized size of antiderivative = 1.00, number
of steps used = 19, number of rules used = 16, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules
used = {6261, 100, 21, 132, 65, 338, 303, 1176, 631, 210, 1179, 642, 95, 218, 212, 209}
\begin {gather*} -2 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\sqrt {2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 65
Rule 95
Rule 100
Rule 132
Rule 209
Rule 210
Rule 212
Rule 218
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{\frac {5}{2} \tanh ^{-1}(a x)}}{x} \, dx &=\int \frac {(1+a x)^{5/4}}{x (1-a x)^{5/4}} \, dx\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {4 \int \frac {-\frac {a}{4}+\frac {a^2 x}{4}}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{a}\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}+\int \frac {(1-a x)^{3/4}}{x (1+a x)^{3/4}} \, dx\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-a \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx+\int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}+4 \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )+4 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+4 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )+2 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}+\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}+\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )-\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )\\ &=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )+\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 93, normalized size = 0.38 \begin {gather*} \frac {4 \left (3+3 a x+3 \sqrt [4]{2} (1+a x)^{3/4} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {1}{2} (1-a x)\right )+(-1+a x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{1+a x}\right )\right )}{3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs.
\(2 (195) = 390\).
time = 0.45, size = 442, normalized size = 1.78 \begin {gather*} 2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {a x + \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {a x - \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (a x + \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1\right )}}{a x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (a x - \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1\right )}}{a x - 1}\right ) + 8 \, \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 2 \, \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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