Optimal. Leaf size=163 \[ \frac {x^2 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^2}-\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^3}-\frac {7 x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{24 a^3}-\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^3}-\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt {a x+1} \sqrt {1-a x}} \]
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Rubi [A] time = 0.55, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6167, 6159, 6129, 90, 80, 50, 41, 216} \[ \frac {x^2 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^2}-\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^3}-\frac {7 x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{24 a^3}-\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^3}-\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt {a x+1} \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 80
Rule 90
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt {1-a x} \sqrt {1+a x} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x^2 (1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2} (-1+2 a x)}{\sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{12 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{8 a^3 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 93, normalized size = 0.57 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (21 \log \left (\sqrt {a^2 x^2-1}+a x\right )+\sqrt {a^2 x^2-1} \left (6 a^3 x^3-16 a^2 x^2+21 a x-32\right )\right )}{24 a^3 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 222, normalized size = 1.36 \[ \left [\frac {2 \, {\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 21 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{48 \, a^{4}}, \frac {{\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{24 \, a^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 128, normalized size = 0.79 \[ \frac {1}{48} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left ({\left (2 \, x {\left (\frac {3 \, x \mathrm {sgn}\relax (x)}{a^{2}} - \frac {8 \, \mathrm {sgn}\relax (x)}{a^{3}}\right )} + \frac {21 \, \mathrm {sgn}\relax (x)}{a^{4}}\right )} x - \frac {32 \, \mathrm {sgn}\relax (x)}{a^{5}}\right )} - \frac {42 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a^{4} {\left | a \right |}} + \frac {{\left (21 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 64 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\relax (x)}{a^{5} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 196, normalized size = 1.20 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-6 x \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4}+16 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3}-27 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2} c +27 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-48 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right )+48 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a c \right )}{24 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \,a^{4}} \]
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 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{a x + 1}\,{d x} \]
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 \[ \int \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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