Optimal. Leaf size=136 \[ \frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 (5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 (15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.19, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6157, 6148, 811, 813, 844, 216, 266, 63, 208} \[ \frac {c^3 (5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 (15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 811
Rule 813
Rule 844
Rule 6148
Rule 6157
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1+a x) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{a^6}\\ &=\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \int \frac {\left (8 a^2+10 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{8 a^6}\\ &=-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 \int \frac {\left (32 a^4+60 a^5 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{32 a^6}\\ &=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \int \frac {-120 a^5+64 a^6 x}{x \sqrt {1-a^2 x^2}} \, dx}{64 a^6}\\ &=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+c^3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (15 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}-\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}+\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{8 a^3}\\ &=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}+\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 0.51 \[ \frac {c^3 \left (\frac {7 \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};a^2 x^2\right )}{x^5}+5 a^5 \left (1-a^2 x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-a^2 x^2\right )\right )}{35 a^6} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 153, normalized size = 1.12 \[ -\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 225 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 120 \, a^{5} c^{3} x^{5} + {\left (120 \, a^{5} c^{3} x^{5} - 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} + 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x - 24 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 385, normalized size = 2.83 \[ -\frac {{\left (6 \, c^{3} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} + \frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} + \frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {15 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} + \frac {\frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} - \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{960 \, a^{4} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 187, normalized size = 1.38 \[ -\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{a}+\frac {c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {15 c^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}+\frac {23 c^{3} \sqrt {-a^{2} x^{2}+1}}{15 a^{2} x}-\frac {11 c^{3} \sqrt {-a^{2} x^{2}+1}}{15 a^{4} x^{3}}-\frac {9 c^{3} \sqrt {-a^{2} x^{2}+1}}{8 x^{2} a^{3}}+\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{5} x^{4}}+\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{5 a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 332, normalized size = 2.44 \[ \frac {c^{3} \arcsin \left (a x\right )}{a} + \frac {3 \, c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {3 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{3}}{2 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x} - \frac {{\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{3}}{a^{4}} + \frac {{\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{x^{4}}\right )} c^{3}}{8 \, a^{5}} + \frac {{\left (\frac {8 \, \sqrt {-a^{2} x^{2} + 1} a^{4}}{x} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{x^{5}}\right )} c^{3}}{15 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 182, normalized size = 1.34 \[ \frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^2\,x}-\frac {9\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^4\,x^3}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.35, size = 687, normalized size = 5.05 \[ a c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - \frac {3 c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {3 c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {3 c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {3 c^{3} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} - \frac {c^{3} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{5}} - \frac {c^{3} \left (\begin {cases} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {otherwise} \end {cases}\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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