Optimal. Leaf size=591 \[ -\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}+\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a} \]
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Rubi [A]
time = 0.30, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6260, 52, 65,
338, 305, 1136, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}-\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}-\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 305
Rule 338
Rule 632
Rule 642
Rule 648
Rule 1136
Rule 1183
Rule 6260
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \tanh ^{-1}(a x)} \, dx &=\int \frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}+\frac {1}{4} \int \frac {1}{\sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {2 \text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-a x}\right )}{a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {2 \text {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {\text {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{\sqrt {2} a}+\frac {\text {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{\sqrt {2} a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}+\frac {\text {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{\sqrt {2} a}-\frac {\text {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{\sqrt {2} a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )} a}-\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )} a}+\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )} a}+\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )} a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{4 a}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}-\frac {\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{4 a}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{4 a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 a}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 a}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 a}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{2 a}\\ &=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}+\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{8 a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 48, normalized size = 0.08 \begin {gather*} \frac {2 e^{\frac {1}{4} \tanh ^{-1}(a x)} \left (-\frac {1}{1+e^{2 \tanh ^{-1}(a x)}}+\, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-e^{2 \tanh ^{-1}(a x)}\right )\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {1}{4}}\, dx\]
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. 2372 vs.
\(2 (443) = 886\).
time = 0.43, size = 2372, normalized size = 4.01 \begin {gather*} \text {Too large to display} \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 [4]{\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 {\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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