Optimal. Leaf size=307 \[ -\frac {(a x+1)^{7/4} (1-a x)^{9/4}}{4 a}-\frac {7 (a x+1)^{3/4} (1-a x)^{9/4}}{24 a}+\frac {7 (a x+1)^{3/4} (1-a x)^{5/4}}{32 a}+\frac {35 (a x+1)^{3/4} \sqrt [4]{1-a x}}{64 a}+\frac {35 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a}-\frac {35 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a}+\frac {35 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a}-\frac {35 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a} \]
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Rubi [A] time = 0.24, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6140, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {(a x+1)^{7/4} (1-a x)^{9/4}}{4 a}-\frac {7 (a x+1)^{3/4} (1-a x)^{9/4}}{24 a}+\frac {7 (a x+1)^{3/4} (1-a x)^{5/4}}{32 a}+\frac {35 (a x+1)^{3/4} \sqrt [4]{1-a x}}{64 a}+\frac {35 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a}-\frac {35 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a}+\frac {35 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a}-\frac {35 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6140
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx &=\int (1-a x)^{5/4} (1+a x)^{7/4} \, dx\\ &=-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {7}{8} \int (1-a x)^{5/4} (1+a x)^{3/4} \, dx\\ &=-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {7}{16} \int \frac {(1-a x)^{5/4}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {35}{64} \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {35}{128} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}-\frac {35 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a}\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}-\frac {35 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a}\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}-\frac {35 \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 )}{64 a}-\frac {35 \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 )}{64 a}\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}-\frac {35 \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 )}{128 a}-\frac {35 \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 )}{128 a}+\frac {35 \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 )}{128 \sqrt {2} a}+\frac {35 \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 )}{128 \sqrt {2} a}\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {35 \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 )}{128 \sqrt {2} a}-\frac {35 \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 )}{128 \sqrt {2} a}-\frac {35 \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 )}{64 \sqrt {2} a}+\frac {35 \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 )}{64 \sqrt {2} a}\\ &=\frac {35 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a}+\frac {7 (1-a x)^{5/4} (1+a x)^{3/4}}{32 a}-\frac {7 (1-a x)^{9/4} (1+a x)^{3/4}}{24 a}-\frac {(1-a x)^{9/4} (1+a x)^{7/4}}{4 a}+\frac {35 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a}-\frac {35 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a}+\frac {35 \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 )}{128 \sqrt {2} a}-\frac {35 \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 )}{128 \sqrt {2} a}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.14 \[ -\frac {8\ 2^{3/4} (1-a x)^{9/4} \, _2F_1\left (-\frac {7}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-a x)\right )}{9 a} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.66, size = 541, normalized size = 1.76 \[ -\frac {420 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) + 420 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) + 105 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 105 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + 4 \, {\left (48 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 118 \, a x - 43\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, a} \]
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 \[ \int {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}\,{d x} \]
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 \[ \int {\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \]
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 \[ \int \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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