Optimal. Leaf size=168 \[ -\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}-\frac {123}{64} a^4 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {123}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6261, 101,
156, 12, 95, 218, 212, 209} \begin {gather*} -\frac {123}{64} a^4 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {123}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 x}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}{32 x^2}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{a x+1}}{8 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 218
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{-\frac {3}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac {(1-a x)^{3/4}}{x^5 (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {1}{4} \int \frac {-\frac {9 a}{2}+3 a^2 x}{x^4 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {1}{12} \int \frac {-\frac {45 a^2}{4}+9 a^3 x}{x^3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {1}{24} \int \frac {-\frac {189 a^3}{8}+\frac {45 a^4 x}{4}}{x^2 \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}-\frac {1}{24} \int -\frac {369 a^4}{16 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}+\frac {1}{128} \left (123 a^4\right ) \int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}+\frac {1}{32} \left (123 a^4\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}-\frac {1}{64} \left (123 a^4\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {1}{64} \left (123 a^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}+\frac {3 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 x^3}-\frac {15 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{32 x^2}+\frac {63 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 x}-\frac {123}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {123}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 86, normalized size = 0.51 \begin {gather*} \frac {(1-a x)^{3/4} \left (-16+8 a x-6 a^2 x^2+33 a^3 x^3+63 a^4 x^4-82 a^4 x^4 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{1+a x}\right )\right )}{64 x^4 (1+a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{5}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 161, normalized size = 0.96 \begin {gather*} -\frac {246 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 123 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 123 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, {\left (63 \, a^{4} x^{4} - 93 \, a^{3} x^{3} + 54 \, a^{2} x^{2} - 40 \, a x + 16\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{128 \, x^{4}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{5} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^5\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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