Optimal. Leaf size=115 \[ \frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}} \]
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Rubi [A]
time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6262, 677, 679,
675, 214} \begin {gather*} \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 677
Rule 679
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{7/2}} \, dx\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac {1}{2} (3 c) \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-3 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+(6 a c) \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 )\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 57, normalized size = 0.50 \begin {gather*} \frac {(1+a x)^{5/2} (c-a c x)^{3/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {1}{2} (1+a x)\right )}{10 a c^2 (1-a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.28, size = 127, normalized size = 1.10
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c a x -2 \sqrt {c}\, \sqrt {\left (a x +1\right ) c}\, a x -3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +4 \sqrt {\left (a x +1\right ) c}\, \sqrt {c}\right )}{c^{\frac {3}{2}} \left (a x -1\right )^{2} \sqrt {\left (a x +1\right ) c}\, a}\) | \(127\) |
risch | \(-\frac {2 \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \left (-\frac {\sqrt {c x a +c}}{2 \left (c x a -c \right )}-\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(170\) |
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 = 0.36, size = 259, normalized size = 2.25 \begin {gather*} \left [-\frac {4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x - 2\right )} - \frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \log \left (-\frac {a^{2} x^{2} + 2 \, a x + \frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt {c}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x - 2\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\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 {\left (a x + 1\right )^{3}}{\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 70, normalized size = 0.61 \begin {gather*} \frac {\frac {3 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, \sqrt {a c x + c} - \frac {2 \, \sqrt {a c x + c} c}{a c x - c}}{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 {{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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