Optimal. Leaf size=146 \[ \frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}} \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 )^{7/2}} \, dx &=\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{7/2} (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{9/2} x} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx}{a}\\ &=\frac {c \text {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {(11 c) \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^3}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^4}\\ &=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 46, normalized size = 0.32 \begin {gather*} \frac {-7 x+\frac {11 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};1-\frac {1}{a x}\right )}{a}}{7 \left (c-\frac {c}{a x}\right )^{7/2}} \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. \(395\) vs.
\(2(124)=248\).
time = 0.77, size = 396, normalized size = 2.71
method | result | size |
risch | \(-\frac {a x -1}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {11 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {102 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{35 a^{8} c \left (x -\frac {1}{a}\right )^{3}}-\frac {712 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{105 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {1516 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{105 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{7 a^{9} c \left (x -\frac {1}{a}\right )^{4}}\right ) a^{3} \sqrt {a c x \left (a x -1\right )}}{c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(293\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2310 a^{\frac {11}{2}} \sqrt {\left (a x -1\right ) x}\, x^{5}-2100 a^{\frac {9}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{3}-11550 x^{4} \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}}+5368 a^{\frac {7}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}+23100 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}+1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{5} x^{5}-4928 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{4}-23100 x^{2} \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}}+11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+1540 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+11550 x \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}-11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}+5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x -2310 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{210 \sqrt {\left (a x -1\right ) x}\, c^{4} \left (a x -1\right )^{5} \sqrt {a}}\) | \(396\) |
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 = 0.35, size = 346, normalized size = 2.37 \begin {gather*} \left [\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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 (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\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^{3} x \sqrt {c - \frac {c}{a x}} - 4 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} + \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\, dx - \int \frac {1}{a c^{3} x \sqrt {c - \frac {c}{a x}} - 4 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} + \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\, 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. 390 vs.
\(2 (124) = 248\).
time = 0.69, size = 390, normalized size = 2.67 \begin {gather*} -\frac {11 \, \log \left (c^{4} {\left | a \right |} \sqrt {{\left | c \right |}}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} + \frac {11 \, \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} {\left | a \right |} + 17 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a \sqrt {c} - 64 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} c {\left | a \right |} + 140 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a c^{\frac {3}{2}} - 196 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} c^{2} {\left | a \right |} + 182 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c^{\frac {5}{2}} - 112 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{3} {\left | a \right |} + 44 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{\frac {7}{2}} - 10 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{4} {\left | a \right |} + a c^{\frac {9}{2}} \right |}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{4}} \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 )}^{7/2}\,\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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