Optimal. Leaf size=189 \[ \frac {13 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-x \left (c-\frac {c}{a x}\right )^{9/2} \]
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Rubi [A] time = 0.26, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6133, 25, 514, 375, 98, 154, 156, 63, 208} \[ \frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}+\frac {13 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-x \left (c-\frac {c}{a x}\right )^{9/2} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 98
Rule 154
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{9/2} (1-a x)}{1+a x} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{11/2} x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{11/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{11/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2} \left (\frac {13 c^2}{2}+\frac {5 c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {2 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2} \left (\frac {91 c^3}{4}-\frac {21 c^3 x}{4 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{7 c}\\ &=\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {4 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2} \left (\frac {455 c^4}{8}-\frac {665 c^4 x}{8 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{35 c}\\ &=\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {8 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}} \left (\frac {1365 c^5}{16}-\frac {5355 c^5 x}{16 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{105 c}\\ &=\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {16 \operatorname {Subst}\left (\int \frac {\frac {1365 c^6}{32}-\frac {12075 c^6 x}{32 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 c}\\ &=\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {\left (13 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}+\frac {\left (64 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x+\left (13 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )-\left (128 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {13 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \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 = 0.35, size = 133, normalized size = 0.70 \[ \frac {c^4 \left (-105 a^4 x^4+6428 a^3 x^3-1196 a^2 x^2+258 a x-30\right ) \sqrt {c-\frac {c}{a x}}}{105 a^4 x^3}+\frac {13 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 342, normalized size = 1.81 \[ \left [\frac {6720 \, \sqrt {2} a^{3} c^{\frac {9}{2}} x^{3} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 1365 \, a^{3} c^{\frac {9}{2}} x^{3} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, a^{4} x^{3}}, \frac {6720 \, \sqrt {2} a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 1365 \, a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, a^{4} x^{3}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 305, normalized size = 1.61 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (-17430 \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, a^{\frac {9}{2}} x^{5}+6720 \sqrt {\frac {1}{a}}\, a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, x^{5}+10920 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, a^{\frac {7}{2}} x^{3}+8715 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{5} a^{4}-10080 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{5} a^{4}-6720 \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {2}\, a^{\frac {7}{2}} x^{5}-1936 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+456 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}-60 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{210 x^{4} a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}} \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
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 \[ -\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {5 c^{4} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {10 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left (- \frac {10 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right )\, dx - \int \frac {5 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \left (- \frac {c^{4} \sqrt {c - \frac {c}{a x}}}{a^{5} x^{5} + a^{4} x^{4}}\right )\, dx - \int \frac {a c^{4} x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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