Optimal. Leaf size=290 \[ -\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{a x+1}}{32 a^4}-\frac {41 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {123 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a^4}+\frac {123 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{a x+1}}{4 a^2} \]
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Rubi [A] time = 0.21, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6126, 100, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac {x^2 (1-a x)^{7/4} \sqrt [4]{a x+1}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{a x+1}}{32 a^4}-\frac {41 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {123 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a^4}+\frac {123 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a^4} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 100
Rule 147
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6126
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx\\ &=-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {\int \frac {x (1-a x)^{3/4} \left (-2+\frac {3 a x}{2}\right )}{(1+a x)^{3/4}} \, dx}{4 a^2}\\ &=-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac {41 \int \frac {(1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx}{64 a^3}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac {123 \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{128 a^3}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac {123 \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a^4}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac {123 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a^4}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac {123 \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^4}+\frac {123 \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^4}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac {123 \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^4}+\frac {123 \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^4}+\frac {123 \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^4}+\frac {123 \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^4}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac {123 \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^4}-\frac {123 \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^4}+\frac {123 \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^4}-\frac {123 \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^4}\\ &=-\frac {41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac {x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac {(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac {123 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \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^4}-\frac {123 \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^4}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 116, normalized size = 0.40 \[ \frac {(1-a x)^{7/4} \left (-7 a^2 x^2 \sqrt [4]{a x+1}+12 \sqrt [4]{2} \, _2F_1\left (-\frac {5}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2} (1-a x)\right )-20 \sqrt [4]{2} \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2} (1-a x)\right )+7 \sqrt [4]{2} \, _2F_1\left (\frac {3}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2} (1-a x)\right )\right )}{28 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.50, size = 553, normalized size = 1.91 \[ \frac {492 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{12} \sqrt {\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - \sqrt {2} a^{12} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - 1\right ) + 492 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{12} \sqrt {-\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - \sqrt {2} a^{12} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} + 1\right ) - 123 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + 123 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (16 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 54 \, a^{2} x^{2} - 93 \, a x + 63\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{256 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{3}}{\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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