Optimal. Leaf size=77 \[ \frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6266, 6263,
272, 43, 52, 65, 214} \begin {gather*} \frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 52
Rule 65
Rule 214
Rule 272
Rule 6263
Rule 6266
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{3 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \text {Subst}\left (\int \frac {\left (1-a^2 x\right )^{3/2}}{x^2} \, dx,x,x^2\right )}{2 a^3}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{4 a}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{2 a^3}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 77, normalized size = 1.00 \begin {gather*} \frac {\left (c^3+\frac {c^3}{2 a^2 x^2}\right ) \sqrt {1-a^2 x^2}}{a}+\frac {3 c^3 \log (a x)}{2 a}-\frac {3 c^3 \log \left (1+\sqrt {1-a^2 x^2}\right )}{2 a} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.22, size = 118, normalized size = 1.53
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) c^{3}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3}}-\frac {\left (-2 \sqrt {-a^{2} x^{2}+1}+3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) c^{3}}{2 a}\) | \(72\) |
default | \(\frac {c^{3} \left (a^{6} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {3 a^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{a^{3}}\) | \(118\) |
meijerg | \(\frac {3 c^{3} \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }\, a}+\frac {c^{3} \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{a \sqrt {\pi }}+\frac {c^{3} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {3 c^{3} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs.
\(2 (65) = 130\).
time = 0.47, size = 188, normalized size = 2.44 \begin {gather*} -a^{3} c^{3} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + \frac {3 \, c^{3} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} - \frac {3 \, c^{3}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {{\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{2 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 78, normalized size = 1.01 \begin {gather*} \frac {3 \, a^{2} c^{3} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 2 \, a^{2} c^{3} x^{2} + {\left (2 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 18.75, size = 104, normalized size = 1.35 \begin {gather*} \frac {2 c^{3} \sqrt {- a^{2} x^{2} + 1} + \frac {3 c^{3} \log {\left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}}{2} - \frac {3 c^{3} \log {\left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}}{2} + \frac {c^{3}}{2 \cdot \left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}}\right )} + \frac {c^{3}}{2 \left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}}\right )}}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 97, normalized size = 1.26 \begin {gather*} -\frac {3 \, a^{4} c^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{4} c^{3} \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3}}{x^{2}}}{4 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 64, normalized size = 0.83 \begin {gather*} \frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^3\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{2\,a}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________