Optimal. Leaf size=75 \[ -\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+c^3 (1-a x) \sqrt {1-a^2 x^2}-c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-c^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.17, antiderivative size = 75, 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 = {6128, 1809, 815, 844, 216, 266, 63, 208} \[ -\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+c^3 (1-a x) \sqrt {1-a^2 x^2}-c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-c^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x} \, dx &=c \int \frac {(c-a c x)^2 \sqrt {1-a^2 x^2}}{x} \, dx\\ &=-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int \frac {\left (-3 a^2 c^2+6 a^3 c^2 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx}{3 a^2}\\ &=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int \frac {6 a^4 c^2-6 a^5 c^2 x}{x \sqrt {1-a^2 x^2}} \, dx}{6 a^4}\\ &=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+c^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (a c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}-c^3 \sin ^{-1}(a x)+\frac {1}{2} c^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}-c^3 \sin ^{-1}(a x)-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}-c^3 \sin ^{-1}(a x)-c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 135, normalized size = 1.80 \[ \frac {c^3 \left (-2 a^4 x^4+6 a^3 x^3-2 a^2 x^2+3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+18 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-6 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-6 a x+4\right )}{6 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 88, normalized size = 1.17 \[ 2 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \frac {1}{3} \, {\left (a^{2} c^{3} x^{2} - 3 \, a c^{3} x + 2 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 95, normalized size = 1.27 \[ -\frac {a c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {a c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, c^{3} + {\left (a^{2} c^{3} x - 3 \, a c^{3}\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 110, normalized size = 1.47 \[ \frac {c^{3} a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}+\frac {2 c^{3} \sqrt {-a^{2} x^{2}+1}}{3}-c^{3} a x \sqrt {-a^{2} x^{2}+1}-\frac {c^{3} a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-c^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 100, normalized size = 1.33 \[ \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{2} - \sqrt {-a^{2} x^{2} + 1} a c^{3} x - c^{3} \arcsin \left (a x\right ) - c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {2}{3} \, \sqrt {-a^{2} x^{2} + 1} c^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 110, normalized size = 1.47 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^4\,c^3}{3\,{\left (-a^2\right )}^{3/2}}+\frac {a^6\,c^3\,x^2}{3\,{\left (-a^2\right )}^{3/2}}-a\,c^3\,x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-c^3\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.54, size = 226, normalized size = 3.01 \[ - a^{4} c^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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