Optimal. Leaf size=275 \[ \frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (a x+1) \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{12 a c^3 (a x+1)^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{16 a c^3 (a x+1)^4 \sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6143, 6140, 44, 207} \[ \frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (a x+1) \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{12 a c^3 (a x+1)^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{16 a c^3 (a x+1)^4 \sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 207
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1-a x)^2 (1+a x)^5} \, dx}{c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{32 (-1+a x)^2}+\frac {1}{4 (1+a x)^5}+\frac {1}{4 (1+a x)^4}+\frac {3}{16 (1+a x)^3}+\frac {1}{8 (1+a x)^2}-\frac {5}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{16 a c^3 (1+a x)^4 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{12 a c^3 (1+a x)^3 \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (5 \sqrt {1-a^2 x^2}\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{32 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{16 a c^3 (1+a x)^4 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{12 a c^3 (1+a x)^3 \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 101, normalized size = 0.37 \[ \frac {\sqrt {1-a^2 x^2} \left (-15 a^4 x^4-45 a^3 x^3-35 a^2 x^2+15 a x+15 (a x-1) (a x+1)^4 \tanh ^{-1}(a x)+32\right )}{96 a c^3 (a x-1) (a x+1)^4 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 559, normalized size = 2.03 \[ \left [\frac {15 \, {\left (a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) - 4 \, {\left (32 \, a^{5} x^{5} + 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} - 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{384 \, {\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {15 \, {\left (a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) - 2 \, {\left (32 \, a^{5} x^{5} + 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} - 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{192 \, {\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 238, normalized size = 0.87 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (15 \ln \left (a x -1\right ) x^{5} a^{5}-15 \ln \left (a x +1\right ) x^{5} a^{5}+45 \ln \left (a x -1\right ) x^{4} a^{4}-45 \ln \left (a x +1\right ) x^{4} a^{4}+30 x^{4} a^{4}+30 \ln \left (a x -1\right ) x^{3} a^{3}-30 a^{3} x^{3} \ln \left (a x +1\right )+90 x^{3} a^{3}-30 \ln \left (a x -1\right ) x^{2} a^{2}+30 \ln \left (a x +1\right ) x^{2} a^{2}+70 a^{2} x^{2}-45 \ln \left (a x -1\right ) x a +45 a x \ln \left (a x +1\right )-30 a x -15 \ln \left (a x -1\right )+15 \ln \left (a x +1\right )-64\right )}{192 \left (a^{2} x^{2}-1\right ) c^{4} a \left (a x -1\right ) \left (a x +1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 122, normalized size = 0.44 \[ -\frac {15 \, a^{4} x^{4} + 45 \, a^{3} x^{3} + 35 \, a^{2} x^{2} - 15 \, a x - 32}{96 \, {\left (a^{6} c^{\frac {7}{2}} x^{5} + 3 \, a^{5} c^{\frac {7}{2}} x^{4} + 2 \, a^{4} c^{\frac {7}{2}} x^{3} - 2 \, a^{3} c^{\frac {7}{2}} x^{2} - 3 \, a^{2} c^{\frac {7}{2}} x - a c^{\frac {7}{2}}\right )}} + \frac {5 \, \log \left (a x + 1\right )}{64 \, a c^{\frac {7}{2}}} - \frac {5 \, \log \left (a x - 1\right )}{64 \, a c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________