Optimal. Leaf size=96 \[ \frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6268, 25, 528,
382, 79, 53, 65, 214} \begin {gather*} -\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 53
Rule 65
Rule 79
Rule 214
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} x} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx}{a}\\ &=\frac {c \text {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {(7 c) \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 55, normalized size = 0.57 \begin {gather*} \frac {x \left (-3 a x+7 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {1}{a x}\right )\right )}{3 c \sqrt {c-\frac {c}{a x}} (-1+a x)} \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. \(259\) vs.
\(2(82)=164\).
time = 0.93, size = 260, normalized size = 2.71
method | result | size |
risch | \(-\frac {a x -1}{a c \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {7 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )^{2}}-\frac {22 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{4} c \left (x -\frac {1}{a}\right )}\right ) a \sqrt {a c x \left (a x -1\right )}}{c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(203\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (42 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}-36 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -126 x^{2} \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}}+21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+28 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+126 x \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}-63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}+63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x -42 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{6 \sqrt {\left (a x -1\right ) x}\, c^{2} \left (a x -1\right )^{3} \sqrt {a}}\) | \(260\) |
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 = 0.38, size = 238, normalized size = 2.48 \begin {gather*} \left [\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\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}{a c x \sqrt {c - \frac {c}{a x}} - 2 c \sqrt {c - \frac {c}{a x}} + \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\, dx - \int \frac {1}{a c x \sqrt {c - \frac {c}{a x}} - 2 c \sqrt {c - \frac {c}{a x}} + \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\, dx \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. 244 vs.
\(2 (82) = 164\).
time = 0.51, size = 244, normalized size = 2.54 \begin {gather*} -\frac {7 \, \log \left (c^{2} {\left | a \right |} \sqrt {{\left | c \right |}}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} + \frac {7 \, \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} {\left | a \right |} + 9 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a \sqrt {c} - 16 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c {\left | a \right |} + 14 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{\frac {3}{2}} - 6 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{2} {\left | a \right |} + a c^{\frac {5}{2}} \right |}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{2}} \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 )}^2}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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