Optimal. Leaf size=102 \[ -\frac {5 a^2 c^3 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 1807, 807, 266, 47, 63, 208} \[ -\frac {5 a^2 c^3 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {5}{8} a^4 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x^5} \, dx &=c \int \frac {(c-a c x)^2 \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} c \int \frac {\left (8 a c^2-5 a^2 c^2 x\right ) \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (5 a^2 c^3\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c^3 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {1}{16} \left (5 a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c^3 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^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 )\\ &=-\frac {5 a^2 c^3 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 a c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 99, normalized size = 0.97 \[ \frac {c^3 \left (16 a^5 x^5+9 a^4 x^4-32 a^3 x^3-3 a^2 x^2+15 a^4 x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+16 a x-6\right )}{24 x^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 84, normalized size = 0.82 \[ -\frac {15 \, a^{4} c^{3} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (16 \, a^{3} c^{3} x^{3} + 9 \, a^{2} c^{3} x^{2} - 16 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 300, normalized size = 2.94 \[ \frac {{\left (3 \, a^{5} c^{3} - \frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{3}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c^{3}}{x^{2}} + \frac {48 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {5 \, a^{5} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {48 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{3} {\left | a \right |}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c^{3} {\left | a \right |}}{x^{2}} - \frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{3} {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 144, normalized size = 1.41 \[ -c^{3} \left (-a^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 a^{3} \sqrt {-a^{2} x^{2}+1}}{x}+2 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}-\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 122, normalized size = 1.20 \[ \frac {5}{8} \, a^{4} c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3}}{3 \, x} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3}}{8 \, x^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a c^{3}}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 113, normalized size = 1.11 \[ \frac {2\,a\,c^3\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {3\,a^2\,c^3\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {2\,a^3\,c^3\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {a^4\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.34, size = 347, normalized size = 3.40 \[ - a^{4} c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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