Optimal. Leaf size=101 \[ \frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {13}{8} c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.25, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {13}{8} c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x} \, dx\\ &=\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {c \int \frac {\sqrt {1-a^2 x^2} \left (-4 a^2 c^3+13 a^3 c^3 x-12 a^4 c^3 x^2\right )}{x} \, dx}{4 a^2}\\ &=-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c \int \frac {\left (12 a^4 c^3-39 a^5 c^3 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx}{12 a^4}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {c \int \frac {-24 a^6 c^3+39 a^7 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{24 a^6}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+c^4 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\frac {1}{8} \left (13 a c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)+\frac {1}{2} c^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)-\frac {c^4 \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}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 142, normalized size = 1.41 \[ \frac {c^4 \left (2 a^5 x^5-8 a^4 x^4+9 a^3 x^3+8 a^2 x^2+4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+34 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-8 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-11 a x\right )}{8 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 95, normalized size = 0.94 \[ \frac {13}{4} \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{8} \, {\left (2 \, a^{3} c^{4} x^{3} - 8 \, a^{2} c^{4} x^{2} + 11 \, a c^{4} x\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 100, normalized size = 0.99 \[ -\frac {13 \, a c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} - \frac {a c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{8} \, {\left (11 \, a c^{4} + 2 \, {\left (a^{3} c^{4} x - 4 \, a^{2} c^{4}\right )} x\right )} \sqrt {-a^{2} x^{2} + 1} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 115, normalized size = 1.14 \[ -\frac {c^{4} a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}-\frac {11 c^{4} a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {13 c^{4} a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}+c^{4} a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-c^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 105, normalized size = 1.04 \[ -\frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{3} + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{2} - \frac {11}{8} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x - \frac {13}{8} \, c^{4} \arcsin \left (a x\right ) - c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 110, normalized size = 1.09 \[ a^2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}-\frac {a^3\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {11\,a\,c^4\,x\,\sqrt {1-a^2\,x^2}}{8}-\frac {13\,a\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 19.06, size = 420, normalized size = 4.16 \[ a^{5} c^{4} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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