Optimal. Leaf size=170 \[ -\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^4 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6160, 6150, 43} \[ -\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^4 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {e^{-3 \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3}{(1+a x)^3} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \left (\frac {1}{a^3}-\frac {1}{a^3 (1+a x)^3}+\frac {3}{a^3 (1+a x)^2}-\frac {3}{a^3 (1+a x)}\right ) \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)^2}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 86, normalized size = 0.51 \[ -\frac {\sqrt {1-a^2 x^2} \left (a^2 x^2-1\right ) \left (-\frac {3}{a^4 (a x+1)}+\frac {1}{2 a^4 (a x+1)^2}-\frac {3 \log (a x+1)}{a^4}+\frac {x}{a^3}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 477, normalized size = 2.81 \[ \left [-\frac {3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) + {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac {6 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) - {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 106, normalized size = 0.62 \[ \frac {\left (-2 x^{3} a^{3}+6 \ln \left (a x +1\right ) x^{2} a^{2}-4 a^{2} x^{2}+12 a x \ln \left (a x +1\right )+4 a x +6 \ln \left (a x +1\right )+5\right ) \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a x +1\right ) a^{4} x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________