Optimal. Leaf size=161 \[ \frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {16 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 161, 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^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}+\frac {5 a x+6}{3 c^2 x^3 \sqrt {1-a^2 x^2}}+\frac {a x+1}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {5 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \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^4 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1+a x}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {6 a^2+5 a^3 x}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {24 a^4+15 a^5 x}{x^4 \sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {\int \frac {-45 a^5-48 a^6 x}{x^3 \sqrt {1-a^2 x^2}} \, dx}{9 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}+\frac {\int \frac {96 a^6+45 a^7 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{18 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {16 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {\left (5 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {16 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {16 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {(5 a) \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^2}\\ &=\frac {1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6+5 a x}{3 c^2 x^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {5 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {16 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 110, normalized size = 0.68 \begin {gather*} \frac {2+a x+11 a^2 x^2-31 a^3 x^3-17 a^4 x^4+32 a^5 x^5-15 a^3 x^3 (-1+a x) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{6 c^2 x^3 (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 260, normalized size = 1.61
method | result | size |
risch | \(\frac {16 a^{4} x^{4}+3 a^{3} x^{3}-14 a^{2} x^{2}-3 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {a^{3} \left (-\frac {9 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{2 a \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{2 a \left (x +\frac {1}{a}\right )}-5 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2 c^{2}}\) | \(236\) |
default | \(\frac {-\frac {a^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 \left (x +\frac {1}{a}\right )}-\frac {8 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-2 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {9 a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{4 \left (x -\frac {1}{a}\right )}+\frac {a^{2} \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{2}+a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{c^{2}}\) | \(260\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 191, normalized size = 1.19 \begin {gather*} -\frac {\frac {15 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c^{2}} - \frac {15 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c^{2}} - \frac {2 \, {\left (15 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{4} + 10 \, {\left (a^{2} x^{2} - 1\right )} a^{4} - 2 \, a^{4}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{2} - {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}}{12 \, a} + \frac {16 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 6 \, a^{2} x^{2} + 1}{3 \, {\left (a^{2} c^{2} x^{5} - c^{2} x^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.47, size = 178, normalized size = 1.11 \begin {gather*} \frac {14 \, a^{6} x^{6} - 14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} + 14 \, a^{3} x^{3} + 15 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (32 \, a^{5} x^{5} - 17 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{3} c^{2} x^{6} - a^{2} c^{2} x^{5} - a c^{2} x^{4} + c^{2} x^{3}\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^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{8} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \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 = 0.91, size = 244, normalized size = 1.52 \begin {gather*} \frac {a^5\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{3\,c^2\,x^3}-\frac {a\,\sqrt {1-a^2\,x^2}}{2\,c^2\,x^2}-\frac {8\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c^2\,x}+\frac {a^4\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {29\,a^4\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________