Optimal. Leaf size=268 \[ -\frac {2 a n \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{a x+1}\right )}{(1-n) \sqrt {1-a^2 x^2}}+\frac {a 2^{\frac {n+1}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{(1-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{x \sqrt {1-a^2 x^2}} \]
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Rubi [C] time = 0.23, antiderivative size = 97, normalized size of antiderivative = 0.36, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 136} \[ \frac {a 2^{\frac {3}{2}-\frac {n}{2}} \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} F_1\left (\frac {n+3}{2};\frac {n-1}{2},2;\frac {n+5}{2};\frac {1}{2} (a x+1),a x+1\right )}{(n+3) \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2^{\frac {3}{2}-\frac {n}{2}} a (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2} F_1\left (\frac {3+n}{2};\frac {1}{2} (-1+n),2;\frac {5+n}{2};\frac {1}{2} (1+a x),1+a x\right )}{(3+n) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 138, normalized size = 0.51 \[ -\frac {c \sqrt {1-a^2 x^2} e^{n \tanh ^{-1}(a x)} \left ((n+1) \sqrt {1-a^2 x^2}+2 a x e^{\tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-e^{2 \tanh ^{-1}(a x)}\right )+2 a n x e^{\tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};e^{2 \tanh ^{-1}(a x)}\right )\right )}{(n+1) x \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{2}}, x\right ) \]
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 [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}}{x^{2}}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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