Optimal. Leaf size=591 \[ -\frac{(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}-\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}-\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2+\sqrt{2}}}\right )}{4 a}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2-\sqrt{2}}}\right )}{4 a}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 a}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 a} \]
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Rubi [A] time = 0.436628, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {6125, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ -\frac{(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}-\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}-\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{8 a}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2+\sqrt{2}}}\right )}{4 a}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2-\sqrt{2}}}\right )}{4 a}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 a}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 6125
Rule 50
Rule 63
Rule 331
Rule 299
Rule 1122
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
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 \operatorname{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 \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 )} \operatorname{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 )} \operatorname{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}} \operatorname{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}} \operatorname{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}} \operatorname{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}} \operatorname{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 )} \operatorname{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 )} \operatorname{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 )} \operatorname{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 )} \operatorname{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 )} \operatorname{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 )} \operatorname{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*}
Mathematica [C] time = 0.0528874, size = 48, normalized size = 0.08 \[ \frac{2 e^{\frac{1}{4} \tanh ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},-e^{2 \tanh ^{-1}(a x)}\right )-\frac{1}{e^{2 \tanh ^{-1}(a x)}+1}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}\, 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 \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24933, size = 6561, normalized size = 11.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx \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 \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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