Optimal. Leaf size=242 \[ -\frac {a^2 \left (n^2+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{a x+1}\right )}{(1-n) \sqrt {c-a^2 c x^2}}-\frac {a n \sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{2 x \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{2 x^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 242, normalized size of antiderivative = 1.00, 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, 151, 12, 131} \[ -\frac {a^2 \left (n^2+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{a x+1}\right )}{(1-n) \sqrt {c-a^2 c x^2}}-\frac {a n \sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{2 x \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{2 x^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 151
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^3 \sqrt {1-a^2 x^2}} \, dx}{\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 {1}{2}+\frac {n}{2}}}{x^3} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x^2 \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 {1}{2}+\frac {n}{2}} \left (-a n-a^2 x\right )}{x^2} \, dx}{2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {a^2 \left (1+n^2\right ) (1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}}}{x} \, dx}{2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x \sqrt {c-a^2 c x^2}}+\frac {\left (a^2 \left (1+n^2\right ) \sqrt {1-a^2 x^2}\right ) \int \frac {(1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}}}{x} \, dx}{2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x \sqrt {c-a^2 c x^2}}-\frac {a^2 \left (1+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{1+a x}\right )}{(1-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 134, normalized size = 0.55 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{\frac {1}{2}-\frac {n}{2}} (a x+1)^{\frac {n-1}{2}} \left (2 a^2 \left (n^2+1\right ) x^2 \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\frac {1-a x}{a x+1}\right )-(n-1) (a x+1) (a n x+1)\right )}{2 (n-1) x^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{5} - c x^{3}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{3} \sqrt {-a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^3\,\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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