Optimal. Leaf size=164 \[ \frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}-\frac {7 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6283, 837, 849,
821, 272, 65, 214} \begin {gather*} -\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}+\frac {24 a x+35}{15 c^3 x^2 \sqrt {1-a^2 x^2}}+\frac {6 a x+7}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a x+1}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {7 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 849
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {1+a x}{x^3 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {\int \frac {7 a^2+6 a^3 x}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {35 a^4+24 a^5 x}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {105 a^6+48 a^7 x}{x^3 \sqrt {1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {\int \frac {-96 a^7-105 a^8 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{30 a^6 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}+\frac {\left (7 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}+\frac {\left (7 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}-\frac {7 \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 c^3}\\ &=\frac {1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {35+24 a x}{15 c^3 x^2 \sqrt {1-a^2 x^2}}-\frac {7 \sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {16 a \sqrt {1-a^2 x^2}}{5 c^3 x}-\frac {7 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 133, normalized size = 0.81 \begin {gather*} \frac {-15-15 a x+176 a^2 x^2+4 a^3 x^3-249 a^4 x^4+9 a^5 x^5+96 a^6 x^6-105 a^2 x^2 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{30 c^3 x^2 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs.
\(2(140)=280\).
time = 0.08, size = 326, normalized size = 1.99
method | result | size |
default | \(-\frac {-\frac {9 a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 \left (x +\frac {1}{a}\right )}+\frac {a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {a \left (-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}\right )}{8}+\frac {7 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {39 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{20 a \left (x -\frac {1}{a}\right )^{3}}-\frac {11 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{10}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}}{c^{3}}\) | \(326\) |
risch | \(\frac {2 a^{3} x^{3}+a^{2} x^{2}-2 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {a^{2} \left (-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{4 a}-\frac {39 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{8 a \left (x -\frac {1}{a}\right )}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{2 a^{2}}+\frac {9 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{8 a \left (x +\frac {1}{a}\right )}-7 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2 c^{3}}\) | \(431\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 241, normalized size = 1.47 \begin {gather*} \frac {116 \, a^{7} x^{7} - 116 \, a^{6} x^{6} - 232 \, a^{5} x^{5} + 232 \, a^{4} x^{4} + 116 \, a^{3} x^{3} - 116 \, a^{2} x^{2} + 105 \, {\left (a^{7} x^{7} - a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{4} x^{4} + a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (96 \, a^{6} x^{6} + 9 \, a^{5} x^{5} - 249 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 176 \, a^{2} x^{2} - 15 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{5} c^{3} x^{7} - a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{4} + a c^{3} x^{3} - c^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{- a^{6} x^{8} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{9} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.92, size = 357, normalized size = 2.18 \begin {gather*} \frac {11\,a^4\,\sqrt {1-a^2\,x^2}}{30\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^4\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,x^2}-\frac {a\,\sqrt {1-a^2\,x^2}}{c^3\,x}-\frac {29\,a^3\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {673\,a^3\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{2\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________