Optimal. Leaf size=159 \[ -\frac {(1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\sqrt {2} (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\sqrt {a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6134, 6129, 102, 21, 105, 54, 215, 93, 206} \[ -\frac {(1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\sqrt {2} (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\sqrt {a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 54
Rule 93
Rule 102
Rule 105
Rule 206
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {(1-a x)^{3/2} \int \frac {e^{-\tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int \frac {x^{3/2}}{(1-a x) \sqrt {1+a x}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \int \frac {-\frac {1}{2}-\frac {a x}{2}}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1-a x)^{3/2} \int \frac {\sqrt {1+a x}}{\sqrt {x} (1-a x)} \, dx}{2 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {(1-a x)^{3/2} \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {\left (2 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {\sqrt {2} (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 105, normalized size = 0.66 \[ \frac {\sqrt {1-a x} \left (\sqrt {a} \sqrt {x} \sqrt {a x+1}+\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )\right )}{a^{3/2} c \sqrt {x} \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 453, normalized size = 2.85 \[ \left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + \sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right )}{4 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - \frac {\sqrt {2} {\left (a c x - c\right )} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}}}{{\left (3 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{2 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
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 = 0.06, size = 168, normalized size = 1.06 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (2 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+2 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} c^{2} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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