Optimal. Leaf size=109 \[ \frac {2 a^2 (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}} \]
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Rubi [A] time = 0.32, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1805, 1807, 807, 266, 63, 208} \[ \frac {2 a^2 (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1805
Rule 1807
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx &=c \int \frac {(1+a x)^2}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\int \frac {-1-2 a x-2 a^2 x^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}+\frac {\int \frac {4 a c+5 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{2 c}\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}+\frac {1}{2} \left (5 a^2\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}+\frac {1}{4} \left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{2 c}\\ &=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}-\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 94, normalized size = 0.86 \[ \frac {\frac {\left (-8 a^2 x^2+3 a x+1\right ) \sqrt {c-a^2 c x^2}}{x^2 (a x-1)}-5 a^2 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+5 a^2 \sqrt {c} \log (x)}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 208, normalized size = 1.91 \[ \left [\frac {5 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (8 \, a^{2} x^{2} - 3 \, a x - 1\right )}}{4 \, {\left (a c x^{3} - c x^{2}\right )}}, -\frac {5 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (8 \, a^{2} x^{2} - 3 \, a x - 1\right )}}{2 \, {\left (a c x^{3} - c x^{2}\right )}}\right ] \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 124, normalized size = 1.14 \[ -\frac {5 a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2 \sqrt {c}}-\frac {2 a \sqrt {-a^{2} c \,x^{2}+c}}{c x}-\frac {2 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} c \,x^{2}+c}}{2 c \,x^{2}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \]
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 \[ -\int \frac {{\left (a\,x+1\right )}^2}{x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{a x^{4} \sqrt {- a^{2} c x^{2} + c} - x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a x^{4} \sqrt {- a^{2} c x^{2} + c} - x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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