Optimal. Leaf size=83 \[ -\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 \text {ArcSin}(a x)+2 a c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 1821,
829, 858, 222, 272, 65, 214} \begin {gather*} -\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 (a x+4) \sqrt {1-a^2 x^2}+2 a c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} a c^3 \text {ArcSin}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 829
Rule 858
Rule 1821
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x^2} \, dx &=c \int \frac {(c-a c x)^2 \sqrt {1-a^2 x^2}}{x^2} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-c \int \frac {\left (2 a c^2+a^2 c^2 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx\\ &=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {c \int \frac {-4 a^3 c^2-a^4 c^2 x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\left (2 a c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\frac {1}{2} \left (a^2 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 \sin ^{-1}(a x)-\left (a c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 \sin ^{-1}(a x)+\frac {\left (2 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 )}{a}\\ &=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 \sin ^{-1}(a x)+2 a c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 143, normalized size = 1.72 \begin {gather*} -\frac {c^3 \left (2+4 a x-3 a^2 x^2-4 a^3 x^3+a^4 x^4+2 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+2 a x \sqrt {1-a^2 x^2} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-4 a x \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{2 x \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.88, size = 107, normalized size = 1.29
method | result | size |
default | \(-c^{3} \left (a^{4} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+2 a \sqrt {-a^{2} x^{2}+1}+\frac {\sqrt {-a^{2} x^{2}+1}}{x}-2 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(107\) |
risch | \(\frac {\left (a^{2} x^{2}-1\right ) c^{3}}{x \sqrt {-a^{2} x^{2}+1}}-\left (-\frac {\sqrt {-a^{2} x^{2}+1}\, a^{2} x}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+2 a \sqrt {-a^{2} x^{2}+1}-2 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) c^{3}\) | \(115\) |
meijerg | \(\frac {a^{2} c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {a \,c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{\sqrt {\pi }}-\frac {a \,c^{3} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{\sqrt {\pi }}-\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{x}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 102, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \frac {1}{2} \, a c^{3} \arcsin \left (a x\right ) + 2 \, a c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} a c^{3} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 104, normalized size = 1.25 \begin {gather*} \frac {2 \, a c^{3} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 4 \, a c^{3} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, a c^{3} x + {\left (a^{2} c^{3} x^{2} - 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.82, size = 199, normalized size = 2.40 \begin {gather*} - a^{4} c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} 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 ) - 2 a 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 ) + 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 ) \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. 152 vs.
\(2 (73) = 146\).
time = 0.45, size = 152, normalized size = 1.83 \begin {gather*} \frac {a^{4} c^{3} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} 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 )}{{\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{2 \, x {\left | a \right |}} + \frac {1}{2} \, {\left (a^{2} c^{3} x - 4 \, a c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 108, normalized size = 1.30 \begin {gather*} \frac {a^2\,c^3\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{x}-2\,a\,c^3\,\sqrt {1-a^2\,x^2}-\frac {a^2\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-a\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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