Optimal. Leaf size=271 \[ -\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}+\frac {a \log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt {2}}-\frac {a \log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt {2}}-\frac {1}{2} a \tanh ^{-1}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6126, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}+\frac {a \log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt {2}}-\frac {a \log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt {2}}-\frac {1}{2} a \tanh ^{-1}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{4} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {\sqrt [8]{1+a x}}{x^2 \sqrt [8]{1-a x}} \, dx\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}+\frac {1}{4} a \int \frac {1}{x \sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}+(2 a) \operatorname {Subst}\left (\int \frac {1}{-1+x^8} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-a \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-a \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} a \tanh ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{4 \sqrt {2}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{4 \sqrt {2}}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} a \tanh ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \log \left (1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {1}{2} a \tanh ^{-1}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \log \left (1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 58, normalized size = 0.21 \[ -\frac {(1-a x)^{7/8} \left (2 a x \, _2F_1\left (\frac {7}{8},1;\frac {15}{8};\frac {1-a x}{a x+1}\right )+7 a x+7\right )}{7 x (a x+1)^{7/8}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.24, size = 544, normalized size = 2.01 \[ -\frac {4 \, a x \arctan \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + 2 \, a x \log \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + 1\right ) - 2 \, a x \log \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} - 1\right ) - 4 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {a^{4} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}}}{a^{4}}\right ) - 4 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (\frac {a^{4} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}}}{a^{4}}\right ) + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (a^{2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}\right ) - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (a^{2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}\right ) - 8 \, {\left (a x - 1\right )} \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}}{8 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {1}{4}}}{x^{2}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt [4]{\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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