Optimal. Leaf size=85 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{\sqrt {2} a c^{5/2}}-\frac {\sqrt {c-a c x}}{a c^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6127, 667, 661, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{\sqrt {2} a c^{5/2}}-\frac {\sqrt {c-a c x}}{a c^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 667
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac {\int \frac {\sqrt {c-a c x}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {\sqrt {c-a c x}}{a c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx}{2 c^2}\\ &=-\frac {\sqrt {c-a c x}}{a c^3 \sqrt {1-a^2 x^2}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )}{c}\\ &=-\frac {\sqrt {c-a c x}}{a c^3 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{\sqrt {2} a c^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 55, normalized size = 0.65 \[ -\frac {(1-a x)^{3/2} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (a x+1)\right )}{a c \sqrt {a x+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 234, normalized size = 2.75 \[ \left [\frac {\sqrt {2} {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{4 \, {\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}\right ] \]
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 [A] time = 0.05, size = 82, normalized size = 0.96 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c \left (a x +1\right )}-2 \sqrt {c}\right )}{2 c^{\frac {7}{2}} \left (a x -1\right ) \left (a x +1\right ) a} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (-a c x + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-a\,c\,x\right )}^{5/2}\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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