Optimal. Leaf size=187 \[ \frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}-\frac {18 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 866,
1819, 1821, 821, 272, 65, 214} \begin {gather*} -\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}+\frac {a^3 (93 a x+80)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 a^3 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {18 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 1821
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 (c-a c x)^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^4 (c-a c x)^4} \, dx\\ &=\frac {\int \frac {(c+a c x)^4}{x^4 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 c^4-20 a c^4 x-35 a^2 c^4 x^2-40 a^3 c^4 x^3-32 a^4 c^4 x^4}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {15 c^4+60 a c^4 x+120 a^2 c^4 x^2+180 a^3 c^4 x^3+144 a^4 c^4 x^4}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {-15 c^4-60 a c^4 x-135 a^2 c^4 x^2-240 a^3 c^4 x^3}{x^4 \sqrt {1-a^2 x^2}} \, dx}{15 c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}+\frac {\int \frac {180 a c^4+435 a^2 c^4 x+720 a^3 c^4 x^2}{x^3 \sqrt {1-a^2 x^2}} \, dx}{45 c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {\int \frac {-870 a^2 c^4-1620 a^3 c^4 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{90 c^7}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}+\frac {\left (18 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}+\frac {\left (9 a^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{c^3}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}-\frac {(18 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 )}{c^3}\\ &=\frac {8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (80+93 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^3 x^3}-\frac {2 a \sqrt {1-a^2 x^2}}{c^3 x^2}-\frac {29 a^2 \sqrt {1-a^2 x^2}}{3 c^3 x}-\frac {18 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 121, normalized size = 0.65 \begin {gather*} \frac {-5-20 a x-85 a^2 x^2+604 a^3 x^3-328 a^4 x^4-578 a^5 x^5+424 a^6 x^6-270 a^3 x^3 (-1+a x)^2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{15 c^3 x^3 (-1+a x)^2 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs.
\(2(163)=326\).
time = 0.78, size = 355, normalized size = 1.90
method | result | size |
risch | \(\frac {29 a^{4} x^{4}+6 a^{3} x^{3}-28 a^{2} x^{2}-6 a x -1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {a^{3} \left (-\frac {2 \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a^{2}}+\frac {\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {16 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-18 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c^{3}}\) | \(331\) |
default | \(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {29 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-7 a^{2} \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )+2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )-4 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+16 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {16 a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c^{3}}\) | \(355\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.35, size = 181, normalized size = 0.97 \begin {gather*} \frac {324 \, a^{6} x^{6} - 972 \, a^{5} x^{5} + 972 \, a^{4} x^{4} - 324 \, a^{3} x^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (424 \, a^{5} x^{5} - 1002 \, a^{4} x^{4} + 674 \, a^{3} x^{3} - 70 \, a^{2} x^{2} - 15 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} + 3 \, a c^{3} x^{4} - c^{3} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {a x}{a^{3} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt {- a^{2} x^{2} + 1} - x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt {- a^{2} x^{2} + 1} - x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 393 vs.
\(2 (163) = 326\).
time = 0.45, size = 393, normalized size = 2.10 \begin {gather*} -\frac {18 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{3} {\left | a \right |}} - \frac {{\left (5 \, a^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {335 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {7559 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}} + \frac {25195 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{4} x^{4}} - \frac {36035 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{6} x^{5}} + \frac {24225 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{8} x^{6}} - \frac {6585 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{10} x^{7}}\right )} a^{6} x^{3}}{120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} - \frac {\frac {117 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{6}}{x} + \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{6}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{6}}{x^{3}}}{24 \, a^{2} c^{9} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 328, normalized size = 1.75 \begin {gather*} \frac {7\,a^5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {4\,a^7\,\sqrt {1-a^2\,x^2}}{15\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{3\,c^3\,x^3}-\frac {2\,a\,\sqrt {1-a^2\,x^2}}{c^3\,x^2}-\frac {29\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c^3\,x}+\frac {93\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\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^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,18{}\mathrm {i}}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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