Optimal. Leaf size=164 \[ \frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {11 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.17, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6268, 25, 528,
382, 100, 159, 162, 65, 214} \begin {gather*} \frac {11 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-x \left (c-\frac {c}{a x}\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 100
Rule 159
Rule 162
Rule 214
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (1-a x)}{1+a x} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{9/2} x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{9/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{9/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2} \left (\frac {11 c^2}{2}+\frac {3 c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2} \left (\frac {55 c^3}{4}-\frac {25 c^3 x}{4 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{5 c}\\ &=\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {4 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}} \left (\frac {165 c^4}{8}-\frac {315 c^4 x}{8 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{15 c}\\ &=\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {8 \text {Subst}\left (\int \frac {\frac {165 c^5}{16}-\frac {795 c^5 x}{16 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 c}\\ &=\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (11 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}+\frac {\left (32 c^4\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\left (11 c^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )-\left (64 c^3\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {11 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 125, normalized size = 0.76 \begin {gather*} \frac {c^3 \sqrt {c-\frac {c}{a x}} \left (6-52 a x+376 a^2 x^2-15 a^3 x^3\right )}{15 a^3 x^2}+\frac {11 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \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. \(280\) vs.
\(2(137)=274\).
time = 0.40, size = 281, normalized size = 1.71
method | result | size |
risch | \(-\frac {\left (15 a^{4} x^{4}-391 a^{3} x^{3}+428 a^{2} x^{2}-58 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} a^{3} \left (a x -1\right )}-\frac {\left (-\frac {11 a^{3} \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 \sqrt {a^{2} c}}-\frac {16 a^{2} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{a^{3} \left (a x -1\right )}\) | \(224\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (480 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{4}-1110 a^{\frac {7}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{4}-480 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{4}+660 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+555 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-720 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-92 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{30 x^{3} a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(281\) |
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 = 0.39, size = 320, normalized size = 1.95 \begin {gather*} \left [\frac {480 \, \sqrt {2} a^{2} c^{\frac {7}{2}} x^{2} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 165 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{30 \, a^{3} x^{2}}, \frac {480 \, \sqrt {2} a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 165 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{15 \, a^{3} x^{2}}\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 \left (- \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left (- \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right )\, dx - \int \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \frac {a c^{3} x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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