Optimal. Leaf size=116 \[ \frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {5}{2} a^2 c^4 (a x+1) \sqrt {1-a^2 x^2}-\frac {5}{2} a^2 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {5}{2} a^2 c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.25, antiderivative size = 116, 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, 1807, 815, 844, 216, 266, 63, 208} \[ \frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {5}{2} a^2 c^4 (a x+1) \sqrt {1-a^2 x^2}-\frac {5}{2} a^2 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {5}{2} a^2 c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1807
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^3} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-\frac {1}{2} c \int \frac {\sqrt {1-a^2 x^2} \left (6 a c^3-5 a^2 c^3 x+2 a^3 c^3 x^2\right )}{x^2} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {1}{2} c \int \frac {\left (5 a^2 c^3+10 a^3 c^3 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx\\ &=\frac {5}{2} a^2 c^4 (1+a x) \sqrt {1-a^2 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {c \int \frac {-10 a^4 c^3-10 a^5 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {5}{2} a^2 c^4 (1+a x) \sqrt {1-a^2 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {1}{2} \left (5 a^2 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} \left (5 a^3 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5}{2} a^2 c^4 (1+a x) \sqrt {1-a^2 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {5}{2} a^2 c^4 \sin ^{-1}(a x)+\frac {1}{4} \left (5 a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {5}{2} a^2 c^4 (1+a x) \sqrt {1-a^2 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {5}{2} a^2 c^4 \sin ^{-1}(a x)-\frac {1}{2} \left (5 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 )\\ &=\frac {5}{2} a^2 c^4 (1+a x) \sqrt {1-a^2 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac {5}{2} a^2 c^4 \sin ^{-1}(a x)-\frac {5}{2} a^2 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 106, normalized size = 0.91 \[ \frac {1}{4} c^4 \left (-10 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+5 a^2 \sin ^{-1}(a x)-10 a^2 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+\frac {2 (a x+1)^2 \left (a^3 x^3-8 a^2 x^2+8 a x-1\right )}{x^2 \sqrt {1-a^2 x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 125, normalized size = 1.08 \[ -\frac {10 \, a^{2} c^{4} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 5 \, a^{2} c^{4} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{2} c^{4} x^{2} + {\left (a^{3} c^{4} x^{3} - 6 \, a^{2} c^{4} x^{2} - 6 \, a c^{4} x + c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 224, normalized size = 1.93 \[ \frac {5 \, a^{3} c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {5 \, a^{3} c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {{\left (a^{3} c^{4} - \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{4}}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} - \frac {1}{2} \, {\left (a^{3} c^{4} x - 6 \, a^{2} c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} + \frac {\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{4} {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 138, normalized size = 1.19 \[ -\frac {c^{4} a^{3} x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {5 c^{4} a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+3 c^{4} a^{2} \sqrt {-a^{2} x^{2}+1}-\frac {5 c^{4} a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 c^{4} a \sqrt {-a^{2} x^{2}+1}}{x}-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 129, normalized size = 1.11 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x + \frac {5}{2} \, a^{2} c^{4} \arcsin \left (a x\right ) - \frac {5}{2} \, a^{2} c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a c^{4}}{x} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 133, normalized size = 1.15 \[ 3\,a^2\,c^4\,\sqrt {1-a^2\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {3\,a\,c^4\,\sqrt {1-a^2\,x^2}}{x}-\frac {a^3\,c^4\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {5\,a^3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {a^2\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.72, size = 357, normalized size = 3.08 \[ a^{5} c^{4} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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