Optimal. Leaf size=222 \[ -\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\frac {\text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a} \]
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Rubi [A]
time = 0.10, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6260, 52, 65,
338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2} a}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2} a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\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 )}{2 \sqrt {2} a}+\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 )}{2 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6260
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \, dx &=\int \frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {1}{2} \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}-\frac {2 \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}-\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {\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 )}{a}-\frac {\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 )}{a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}-\frac {\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 )}{2 a}-\frac {\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 )}{2 a}-\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 )}{2 \sqrt {2} a}-\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 )}{2 \sqrt {2} a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}-\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 )}{2 \sqrt {2} a}+\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 )}{2 \sqrt {2} a}-\frac {\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} a}+\frac {\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} a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\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 )}{2 \sqrt {2} a}+\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 )}{2 \sqrt {2} a}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 149, normalized size = 0.67 \begin {gather*} \frac {-\frac {8 e^{\frac {1}{2} \tanh ^{-1}(a x)}}{1+e^{2 \tanh ^{-1}(a x)}}-2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}\right )+2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}\right )-\sqrt {2} \log \left (1-\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}+e^{\tanh ^{-1}(a x)}\right )+\sqrt {2} \log \left (1+\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}+e^{\tanh ^{-1}(a x)}\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, dx\]
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. 511 vs.
\(2 (174) = 348\).
time = 0.39, size = 511, normalized size = 2.30 \begin {gather*} -\frac {4 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - 1\right ) + 4 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {-\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + 1\right ) - \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a} \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 \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, 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 \sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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