Optimal. Leaf size=106 \[ \frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+3 a c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {1}{2} a c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.24, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6128, 1807, 1809, 815, 844, 216, 266, 63, 208} \[ \frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+3 a c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {1}{2} a c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1807
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^2} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x^2} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-c \int \frac {\sqrt {1-a^2 x^2} \left (3 a c^3-a^2 c^3 x+a^3 c^3 x^2\right )}{x} \, dx\\ &=\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {c \int \frac {\left (-9 a^3 c^3+3 a^4 c^3 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx}{3 a^2}\\ &=-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {c \int \frac {18 a^5 c^3-3 a^6 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{6 a^4}\\ &=-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\left (3 a c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} \left (a^2 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {1}{2} a c^4 \sin ^{-1}(a x)-\frac {1}{2} \left (3 a c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {1}{2} a c^4 (6-a x) \sqrt {1-a^2 x^2}+\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {1}{2} a c^4 \sin ^{-1}(a x)+\frac {\left (3 c^4\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^4 (6-a x) \sqrt {1-a^2 x^2}+\frac {1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {1}{2} a c^4 \sin ^{-1}(a x)+3 a c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 152, normalized size = 1.43 \[ -\frac {c^4 \left (-2 a^5 x^5+9 a^4 x^4-14 a^3 x^3-15 a^2 x^2+9 a x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+24 a x \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-18 a x \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+16 a x+6\right )}{6 x \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 116, normalized size = 1.09 \[ -\frac {6 \, a c^{4} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 18 \, a c^{4} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 16 \, a c^{4} x + {\left (2 \, a^{3} c^{4} x^{3} - 9 \, a^{2} c^{4} x^{2} + 16 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 164, normalized size = 1.55 \[ \frac {a^{4} c^{4} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} + \frac {a^{2} c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {3 \, a^{2} c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{2 \, x {\left | a \right |}} - \frac {1}{6} \, {\left (16 \, a c^{4} + {\left (2 \, a^{3} c^{4} x - 9 \, a^{2} c^{4}\right )} x\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 136, normalized size = 1.28 \[ -\frac {c^{4} a^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {8 c^{4} a \sqrt {-a^{2} x^{2}+1}}{3}+\frac {3 c^{4} a^{2} x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c^{4} a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+3 c^{4} a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 125, normalized size = 1.18 \[ -\frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} + \frac {3}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x + \frac {1}{2} \, a c^{4} \arcsin \left (a x\right ) + 3 \, a c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {8}{3} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 131, normalized size = 1.24 \[ \frac {3\,a^2\,c^4\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{x}-\frac {8\,a\,c^4\,\sqrt {1-a^2\,x^2}}{3}+\frac {a^2\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {a^3\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{3}-a\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.68, size = 306, normalized size = 2.89 \[ a^{5} c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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