Optimal. Leaf size=213 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{a x+1}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (a x+1)}-\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
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Rubi [A] time = 0.45, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{a x+1}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (a x+1)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 149
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^{3/2} (1+a x)^{3/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {(1-a x)^{5/2} \sqrt {1+a x}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {(1-a x)^{3/2} \left (-2 a-3 a^2 x\right )}{x^2 \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1+a x}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {\sqrt {1-a x} \left (a^2+5 a^3 x\right )}{x \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1+a x}-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {a^3+4 a^4 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1+a x}-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}-\frac {\left (a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1+a x}-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}+\frac {\left (a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1+a x}-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)}{2 (1+a x)}-\frac {2 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \sin ^{-1}(a x)}{(1-a x)^{3/2} (1+a x)^{3/2}}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 115, normalized size = 0.54 \[ \frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (2 a^2 x^2+4 a x-1\right )-4 a^2 x^2 \log \left (\sqrt {a^2 x^2-1}+a x\right )+a^2 x^2 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{2 a^2 x \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 316, normalized size = 1.48 \[ \left [\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + a \sqrt {-c} c x \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (2 \, a^{2} c x^{2} + 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, \frac {a c^{\frac {3}{2}} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (2 \, a^{2} c x^{2} + 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 266, normalized size = 1.25 \[ -{\left (\frac {c^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} - \frac {2 \, c^{\frac {3}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {\sqrt {a^{2} c x^{2} - c} c \mathrm {sgn}\relax (x)}{a^{2}} - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{2} {\left | a \right |} \mathrm {sgn}\relax (x) + 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\relax (x) + 4 \, a c^{\frac {7}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 454, normalized size = 2.13 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} x \left (12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{5} c -12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x \,a^{5}-4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c +\sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c +6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{3} c^{2}+3 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}-18 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{3} c^{2}+18 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{2} a -6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{2} a -3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2} c^{2}-3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{2} c^{3}\right )}{6 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} c} \]
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 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.21, size = 376, normalized size = 1.77 \[ c \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) - \frac {2 c \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} + \frac {c \left (\begin {cases} \frac {i a \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {i \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{2 a^{2} x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {\sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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