Optimal. Leaf size=357 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 357, 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 = {6197, 6193, 88} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^7}{(-1+a x)^2 (1+a x)^5} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^7}+\frac {1}{32 a^7 (-1+a x)^2}+\frac {9}{64 a^7 (-1+a x)}-\frac {1}{4 a^7 (1+a x)^5}+\frac {3}{2 a^7 (1+a x)^4}-\frac {59}{16 a^7 (1+a x)^3}+\frac {75}{16 a^7 (1+a x)^2}-\frac {201}{64 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^4}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^3}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 105, normalized size = 0.29 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (2 \left (32 a x+\frac {1}{1-a x}-\frac {150}{a x+1}+\frac {59}{(a x+1)^2}-\frac {16}{(a x+1)^3}+\frac {2}{(a x+1)^4}\right )+9 \log (1-a x)-201 \log (a x+1)\right )}{64 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 208, normalized size = 0.58 \[ \frac {{\left (64 \, a^{6} x^{6} + 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} - 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} + 414 \, a x - 201 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 9 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} \]
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 = 0.07, size = 247, normalized size = 0.69 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (a x -1\right ) \left (64 x^{6} a^{6}+9 \ln \left (a x -1\right ) x^{5} a^{5}-201 \ln \left (a x +1\right ) x^{5} a^{5}+192 x^{5} a^{5}+27 \ln \left (a x -1\right ) x^{4} a^{4}-603 \ln \left (a x +1\right ) x^{4} a^{4}-174 x^{4} a^{4}+18 \ln \left (a x -1\right ) x^{3} a^{3}-402 a^{3} x^{3} \ln \left (a x +1\right )-618 x^{3} a^{3}-18 \ln \left (a x -1\right ) x^{2} a^{2}+402 \ln \left (a x +1\right ) x^{2} a^{2}-118 a^{2} x^{2}-27 \ln \left (a x -1\right ) x a +603 a x \ln \left (a x +1\right )+414 a x -9 \ln \left (a x -1\right )+201 \ln \left (a x +1\right )+208\right )}{64 a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________