Optimal. Leaf size=429 \[ -\frac{(a x+1)^{3/4} (1-a x)^{5/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{3 (a x+1)^{3/4} \sqrt [4]{1-a x} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.257195, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {6143, 6140, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(a x+1)^{3/4} (1-a x)^{5/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{3 (a x+1)^{3/4} \sqrt [4]{1-a x} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{1}{2} \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{\frac{1}{2} \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \sqrt [4]{1-a x} (1+a x)^{3/4} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx}{4 \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{8 \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{8 a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{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 )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{(1-a x)^{5/4} (1+a x)^{3/4} \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-a^2 c x^2} \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 )}{8 \sqrt{2} a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0235308, size = 71, normalized size = 0.17 \[ -\frac{4\ 2^{3/4} (1-a x)^{5/4} \sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{5}{4},\frac{9}{4},\frac{1}{2} (1-a x)\right )}{5 a \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.203, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}\sqrt{-{a}^{2}c{x}^{2}+c}\, 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 \sqrt{-a^{2} c x^{2} + c} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}\, 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 \sqrt{-a^{2} c x^{2} + c} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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