Optimal. Leaf size=139 \[ \frac {17}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {23 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{12 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6126, 99, 151, 12, 93, 298, 203, 206} \[ -\frac {23 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x}+\frac {17}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{12 x^2}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 298
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{\frac {3}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+a x)^{3/4}}{x^4 (1-a x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}+\frac {1}{3} \int \frac {\frac {7 a}{2}+2 a^2 x}{x^3 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {1}{6} \int \frac {-\frac {23 a^2}{4}-\frac {7 a^3 x}{2}}{x^2 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{6} \int \frac {51 a^3}{8 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{16} \left (17 a^3\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{4} \left (17 a^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}-\frac {1}{8} \left (17 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {1}{8} \left (17 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {17}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 78, normalized size = 0.56 \[ -\frac {\sqrt [4]{1-a x} \left (102 a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{a x+1}\right )+23 a^3 x^3+37 a^2 x^2+22 a x+8\right )}{24 x^3 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.68, size = 157, normalized size = 1.13 \[ \frac {102 \, a^{3} x^{3} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (23 \, a^{2} x^{2} + 14 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}} \]
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 = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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