Optimal. Leaf size=110 \[ a^4 c^4 \sin ^{-1}(a x)-\frac {11 a^2 c^4 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \sqrt {1-a^2 x^2}}{4 x^4}+\frac {a c^4 \sqrt {1-a^2 x^2}}{x^3}+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 116, normalized size of antiderivative = 1.05, 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, 811, 844, 216, 266, 63, 208} \[ \frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a^4 c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 811
Rule 844
Rule 1807
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^5} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} c \int \frac {\sqrt {1-a^2 x^2} \left (12 a c^3-13 a^2 c^3 x+4 a^3 c^3 x^2\right )}{x^4} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+\frac {1}{12} c \int \frac {\left (39 a^2 c^3-12 a^3 c^3 x\right ) \sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac {1}{48} c \int \frac {78 a^4 c^3-48 a^5 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac {1}{8} \left (13 a^4 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\left (a^5 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)-\frac {1}{16} \left (13 a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac {1}{8} \left (13 a^2 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 {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 125, normalized size = 1.14 \[ \frac {1}{16} c^4 \left (-13 a^4 \sin ^{-1}(a x)+26 a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \left (-11 a^4 x^4+8 a^3 x^3+9 a^2 x^2+29 a^4 x^4 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-8 a x+2\right )}{x^4 \sqrt {1-a^2 x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 106, normalized size = 0.96 \[ -\frac {16 \, a^{4} c^{4} x^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 13 \, a^{4} c^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (11 \, a^{2} c^{4} x^{2} - 8 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 316, normalized size = 2.87 \[ \frac {a^{5} c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {13 \, a^{5} c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {{\left (a^{5} c^{4} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{4}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c^{4}}{x^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a x^{3}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{4} {\left | a \right |}}{x} - \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c^{4} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{4} {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 118, normalized size = 1.07 \[ \frac {c^{4} a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {13 c^{4} a^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}+\frac {a \,c^{4} \sqrt {-a^{2} x^{2}+1}}{x^{3}}-\frac {11 a^{2} c^{4} \sqrt {-a^{2} x^{2}+1}}{8 x^{2}}-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 109, normalized size = 0.99 \[ a^{4} c^{4} \arcsin \left (a x\right ) + \frac {13}{8} \, a^{4} c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{4}}{x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 113, normalized size = 1.03 \[ \frac {a\,c^4\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {11\,a^2\,c^4\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {a^4\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.36, size = 505, normalized size = 4.59 \[ a^{5} 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^{4} 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 ) + 2 a^{3} 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 ) + 2 a^{2} 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 ) - 3 a c^{4} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \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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