Optimal. Leaf size=255 \[ -\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a^2}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 a^2}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a^2}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt {2} a^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt {2} a^2} \]
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Rubi [A] time = 0.17, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6126, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a^2}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 a^2}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a^2}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt {2} a^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt {2} a^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} x \, dx &=\int \frac {x \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx\\ &=-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac {\int \frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx}{4 a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac {\int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{8 a}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a^2}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a^2}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac {\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^2}-\frac {\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^2}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac {\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^2}-\frac {\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^2}-\frac {\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^2}-\frac {\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^2}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\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 )}{8 \sqrt {2} a^2}+\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 )}{8 \sqrt {2} a^2}-\frac {\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^2}+\frac {\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^2}\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a^2}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a^2}-\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 )}{8 \sqrt {2} a^2}+\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 )}{8 \sqrt {2} a^2}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.22 \[ -\frac {(1-a x)^{3/4} \left (2 \sqrt [4]{2} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-a x)\right )+3 (a x+1)^{5/4}\right )}{6 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.50, size = 535, normalized size = 2.10 \[ -\frac {4 \, \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{6} \sqrt {\frac {\sqrt {2} {\left (a^{3} x - a^{2}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {1}{4}} + {\left (a^{5} x - a^{4}\right )} \sqrt {\frac {1}{a^{8}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {3}{4}} - \sqrt {2} a^{6} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {3}{4}} - 1\right ) + 4 \, \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{6} \sqrt {-\frac {\sqrt {2} {\left (a^{3} x - a^{2}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {1}{4}} - {\left (a^{5} x - a^{4}\right )} \sqrt {\frac {1}{a^{8}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {3}{4}} - \sqrt {2} a^{6} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {3}{4}} + 1\right ) - \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{3} x - a^{2}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {1}{4}} + {\left (a^{5} x - a^{4}\right )} \sqrt {\frac {1}{a^{8}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{3} x - a^{2}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{8}}^{\frac {1}{4}} - {\left (a^{5} x - a^{4}\right )} \sqrt {\frac {1}{a^{8}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (2 \, a^{2} x^{2} + a x - 3\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}} \]
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 \[ \int x \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}\,{d x} \]
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 \[ \int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x\, dx \]
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 \[ \int x \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}\,{d x} \]
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 \[ \int x\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \]
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 \[ \int x \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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