Optimal. Leaf size=267 \[ -\frac {\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2^{\frac {n-1}{2}} n \left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 (3-n) x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac {n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 \left (1-n^2\right ) x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6160, 6150, 100, 145, 69} \[ -\frac {2^{\frac {n-1}{2}} n \left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 (3-n) x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac {n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 \left (1-n^2\right ) x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 145
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2} \int x^3 (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {\left (1-a^2 x^2\right )^{3/2} \int x (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} (-2-a n x) \, dx}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (n \left (1-a^2 x^2\right )^{3/2}\right ) \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {2^{\frac {1}{2} (-1+n)} n (1-a x)^{\frac {3-n}{2}} \left (1-a^2 x^2\right )^{3/2} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 (3-n) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 186, normalized size = 0.70 \[ \frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac {1}{2} (-n-1)} \left (-4 a^4 x^2 (a x+1)^{\frac {n-1}{2}}+\frac {a^2 2^{\frac {n+3}{2}} n (a x-1)^2 \, _2F_1\left (\frac {3}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{n-3}+\frac {4 a^2 \left (n^2 (2 a x-1)+n (3 a x-2)-2\right ) (a x+1)^{\frac {n-1}{2}}}{n^2-1}\right )}{4 a^6 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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