Optimal. Leaf size=243 \[ \frac {2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};\frac {a x+1}{1-a x}\right )}{c (1-n) \sqrt {c-a^2 c x^2}}-\frac {(n+2) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c (n+1) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.32, antiderivative size = 247, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6153, 6150, 129, 155, 12, 131} \[ -\frac {2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {1-a x}{a x+1}\right )}{c (3-n) \sqrt {c-a^2 c x^2}}-\frac {(n+2) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c (n+1) \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 129
Rule 131
Rule 155
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \left (-a (1+n)-a^2 x\right )}{x} \, dx}{a c (1+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(2+n) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {a^2 (1-n) (1+n) (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{a^2 c (1-n) (1+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(2+n) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(2+n) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {1-a x}{1+a x}\right )}{c (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 149, normalized size = 0.61 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}} \left (2 \left (n^2-1\right ) (a x-1)^2 \, _2F_1\left (1,\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1-a x}{a x+1}\right )-(n-3) (a x+1) (n (a x-2)+2 a x-1)\right )}{c (n-3) (n-1) (n+1) \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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