Optimal. Leaf size=122 \[ \frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c (c-a c x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{8 \sqrt {2} a c^{5/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6262, 677, 687,
675, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{8 \sqrt {2} a c^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c (c-a c x)^{3/2}}+\frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 677
Rule 687
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{7/2}} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}}-\frac {\int \frac {1}{(c-a c x)^{3/2} \sqrt {1-a^2 x^2}} \, dx}{4 c}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c (c-a c x)^{3/2}}-\frac {\int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx}{16 c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c (c-a c x)^{3/2}}+\frac {a \text {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )}{8 c}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a (c-a c x)^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c (c-a c x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{8 \sqrt {2} a c^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 57, normalized size = 0.47 \begin {gather*} \frac {(1+a x)^{3/2} \sqrt {c-a c x} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {1}{2} (1+a x)\right )}{12 a c^3 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 156, normalized size = 1.28
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}-2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c a x -2 \sqrt {c}\, \sqrt {\left (a x +1\right ) c}\, a x +\sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -6 \sqrt {\left (a x +1\right ) c}\, \sqrt {c}\right )}{16 c^{\frac {7}{2}} \left (a x -1\right )^{3} \sqrt {\left (a x +1\right ) c}\, a}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 308, normalized size = 2.52 \begin {gather*} \left [\frac {\sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 3\right )}}{32 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 3\right )}}{16 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + 1}{\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 76, normalized size = 0.62 \begin {gather*} \frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {2 \, {\left ({\left (a c x + c\right )}^{\frac {3}{2}} + 2 \, \sqrt {a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c}}{16 \, a {\left | c \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x+1}{\sqrt {1-a^2\,x^2}\,{\left (c-a\,c\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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