Optimal. Leaf size=196 \[ \frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6288, 6285,
105, 157, 162, 65, 214, 212} \begin {gather*} \frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 157
Rule 162
Rule 212
Rule 214
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx &=\frac {\sqrt [4]{1-a^2 x^2} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{5/4}} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2} \int \frac {1}{x^2 (1-a x)^{3/2} (1+a x)} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \int \frac {-\frac {a}{2}-\frac {3 a^2 x}{2}}{x (1-a x)^{3/2} (1+a x)} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \int \frac {\frac {a^2}{2}+a^3 x}{x \sqrt {1-a x} (1+a x)} \, dx}{a c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {\left (a \sqrt [4]{1-a^2 x^2}\right ) \int \frac {1}{x \sqrt {1-a x}} \, dx}{2 c \sqrt [4]{c-a^2 c x^2}}+\frac {\left (a^2 \sqrt [4]{1-a^2 x^2}\right ) \int \frac {1}{\sqrt {1-a x} (1+a x)} \, dx}{2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {\left (a \sqrt [4]{1-a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 86, normalized size = 0.44 \begin {gather*} \frac {\sqrt [4]{1-a^2 x^2} \left (-1+a x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )+a x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-a x\right )\right )}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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