Optimal. Leaf size=269 \[ \frac {2^{\frac {n+1}{2}} n \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{\left (n^2-4 n+3\right ) \sqrt {1-a^2 x^2}}+\frac {2 \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};\frac {a x+1}{1-a x}\right )}{(1-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {3-n}{2}}}{(1-n) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6153, 6150, 105, 69, 131} \[ \frac {2 \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1}{2} (-n-1)} \, _2F_1\left (1,\frac {1}{2} (-n-1);\frac {1-n}{2};\frac {1-a x}{a x+1}\right )}{(n+1) \sqrt {1-a^2 x^2}}-\frac {2^{\frac {n+3}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {1}{2} (-n-1)} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {1}{2} (-n-1);\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{(n+1) \sqrt {1-a^2 x^2}}+\frac {2^{\frac {n+3}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {1-n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{(1-n) \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 69
Rule 105
Rule 131
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x} \, 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} \, 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} \, dx}{\sqrt {1-a^2 x^2}}-\frac {\left (a \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2^{\frac {3+n}{2}} (1-a x)^{\frac {1-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\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} \int \frac {(1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}}}{x} \, dx}{\sqrt {1-a^2 x^2}}-\frac {\left (a \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1+n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (1,\frac {1}{2} (-1-n);\frac {1-n}{2};\frac {1-a x}{1+a x}\right )}{(1+n) \sqrt {1-a^2 x^2}}-\frac {2^{\frac {3+n}{2}} (1-a x)^{\frac {1}{2} (-1-n)} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\frac {1}{2} (-1-n);\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{(1+n) \sqrt {1-a^2 x^2}}+\frac {2^{\frac {3+n}{2}} (1-a x)^{\frac {1-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\frac {1-n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{(1-n) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 207, normalized size = 0.77 \[ \frac {2 \sqrt {c-a^2 c x^2} (1-a x)^{\frac {1}{2} (-n-1)} \left ((n-1) (a x+1)^{\frac {n+1}{2}} \, _2F_1\left (1,-\frac {n}{2}-\frac {1}{2};\frac {1}{2}-\frac {n}{2};\frac {1-a x}{a x+1}\right )+2^{\frac {n+1}{2}} \left ((n+1) (a x-1) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )-(n-1) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},-\frac {n}{2}-\frac {1}{2};\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )\right )\right )}{\left (n^2-1\right ) \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x}, x\right ) \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}}{x}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2}}{x} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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