Optimal. Leaf size=83 \[ -\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 (a x+4) \sqrt {1-a^2 x^2}+2 a c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} a c^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.17, 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 = {6128, 1807, 815, 844, 216, 266, 63, 208} \[ -\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 (a x+4) \sqrt {1-a^2 x^2}+2 a c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} a c^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1807
Rule 6128
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 ) \operatorname {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 ) \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}\\ &=-\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 = 0.11, size = 143, normalized size = 1.72 \[ -\frac {c^3 \left (a^4 x^4-4 a^3 x^3-3 a^2 x^2+2 a x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+2 a x \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-4 a x \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+4 a x+2\right )}{2 x \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 104, normalized size = 1.25 \[ \frac {2 \, a c^{3} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 4 \, a c^{3} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, a c^{3} x + {\left (a^{2} c^{3} x^{2} - 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 152, normalized size = 1.83 \[ \frac {a^{4} c^{3} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{2 \, x {\left | a \right |}} + \frac {1}{2} \, {\left (a^{2} c^{3} x - 4 \, a c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 113, normalized size = 1.36 \[ \frac {c^{3} a^{2} x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {c^{3} a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-2 c^{3} a \sqrt {-a^{2} x^{2}+1}+2 c^{3} a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 102, normalized size = 1.23 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \frac {1}{2} \, a c^{3} \arcsin \left (a x\right ) + 2 \, a c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} a c^{3} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{x} \]
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 \[ \frac {a^2\,c^3\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{x}-2\,a\,c^3\,\sqrt {1-a^2\,x^2}-\frac {a^2\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-a\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.72, size = 199, normalized size = 2.40 \[ - a^{4} c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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