Optimal. Leaf size=76 \[ -\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+a^2 c \text {ArcSin}(a x)-\frac {7}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6283, 1821,
858, 222, 272, 65, 214} \begin {gather*} a^2 c \text {ArcSin}(a x)-\frac {3 a c \sqrt {1-a^2 x^2}}{x}-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {7}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1821
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^3} \, dx &=c \int \frac {(1+a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{2} c \int \frac {-6 a-7 a^2 x-2 a^3 x^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+\frac {1}{2} c \int \frac {7 a^2+2 a^3 x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+\frac {1}{2} \left (7 a^2 c\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\left (a^3 c\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)+\frac {1}{4} \left (7 a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)-\frac {1}{2} (7 c) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {3 a c \sqrt {1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)-\frac {7}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 60, normalized size = 0.79 \begin {gather*} \frac {1}{2} c \left (-\frac {(1+6 a x) \sqrt {1-a^2 x^2}}{x^2}+2 a^2 \text {ArcSin}(a x)-7 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs.
\(2(66)=132\).
time = 0.08, size = 181, normalized size = 2.38
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}+a^{2} x^{2}-6 a x -1\right ) c}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\left (\frac {a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {7 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right ) c\) | \(91\) |
default | \(-c \left (a^{5} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {3 a^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{3} x}{\sqrt {-a^{2} x^{2}+1}}-3 a \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {7 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}+\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}\right )\) | \(181\) |
meijerg | \(\frac {2 a^{2} c \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {a^{2} c \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{\sqrt {\pi }}+\frac {a^{3} c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {2 a^{3} c x}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}-\frac {3 a c \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}\) | \(334\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 116, normalized size = 1.53 \begin {gather*} \frac {3 \, a^{3} c x}{\sqrt {-a^{2} x^{2} + 1}} + a^{2} c \arcsin \left (a x\right ) - \frac {7}{2} \, a^{2} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {a^{2} c}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {c}{2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 85, normalized size = 1.12 \begin {gather*} -\frac {4 \, a^{2} c x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 7 \, a^{2} c x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (6 \, a c x + c\right )}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 6.19, size = 221, normalized size = 2.91 \begin {gather*} a^{3} c \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^{2} c \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 \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 \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \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 \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs.
\(2 (66) = 132\).
time = 0.43, size = 179, normalized size = 2.36 \begin {gather*} \frac {a^{3} c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {{\left (a^{3} c + \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} - \frac {7 \, a^{3} c \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 {\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 83, normalized size = 1.09 \begin {gather*} \frac {a^3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {3\,a\,c\,\sqrt {1-a^2\,x^2}}{x}-\frac {c\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {a^2\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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