Optimal. Leaf size=160 \[ \frac {2^{\frac {n}{2}-1} \left (n^2+2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (2-n)}+\frac {(a x+1)^{n/2} \left (-a n^2 x+n^2+n+2\right ) (1-a x)^{-n/2}}{2 a^4 c n}-\frac {x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{2 a^2 c} \]
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Rubi [A] time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6150, 100, 143, 69} \[ \frac {2^{\frac {n}{2}-1} \left (n^2+2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (2-n)}+\frac {(a x+1)^{n/2} \left (-a n^2 x+n^2+n+2\right ) (1-a x)^{-n/2}}{2 a^4 c n}-\frac {x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{2 a^2 c} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 143
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{c-a^2 c x^2} \, dx &=\frac {\int x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \, dx}{c}\\ &=-\frac {x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{2 a^2 c}-\frac {\int x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} (-2-a n x) \, dx}{2 a^2 c}\\ &=-\frac {x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{2 a^2 c}+\frac {(1-a x)^{-n/2} (1+a x)^{n/2} \left (2+n+n^2-a n^2 x\right )}{2 a^4 c n}-\frac {\left (2+n^2\right ) \int (1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}} \, dx}{2 a^3 c}\\ &=-\frac {x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{2 a^2 c}+\frac {(1-a x)^{-n/2} (1+a x)^{n/2} \left (2+n+n^2-a n^2 x\right )}{2 a^4 c n}+\frac {2^{-1+\frac {n}{2}} \left (2+n^2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 120, normalized size = 0.75 \[ \frac {(1-a x)^{-n/2} \left (2^{n/2} n \left (n^2+2\right ) (a x-1) \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )-(n-2) (a x+1)^{n/2} \left (n \left (a^2 x^2-1\right )+n^2 (a x-1)-2\right )\right )}{2 a^4 c (n-2) n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}, x\right ) \]
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 \[ \int -\frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{3}}{-a^{2} c \,x^{2}+c}\, dx \]
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 \[ -\int \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \]
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 \[ - \frac {\int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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