Optimal. Leaf size=155 \[ -\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} \left (2 a n x+n^2+6\right ) (1-a x)^{1-\frac {n}{2}}}{24 a^4}-\frac {x^2 (a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{4 a^2} \]
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Rubi [A] time = 0.12, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6126, 100, 147, 69} \[ -\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} \left (2 a n x+n^2+6\right ) (1-a x)^{1-\frac {n}{2}}}{24 a^4}-\frac {x^2 (a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rule 6126
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} x^3 \, dx &=\int x^3 (1-a x)^{-n/2} (1+a x)^{n/2} \, dx\\ &=-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {\int x (1-a x)^{-n/2} (1+a x)^{n/2} (-2-a n x) \, dx}{4 a^2}\\ &=-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}} \left (6+n^2+2 a n x\right )}{24 a^4}+\frac {\left (n \left (8+n^2\right )\right ) \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx}{24 a^3}\\ &=-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}} \left (6+n^2+2 a n x\right )}{24 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 182, normalized size = 1.17 \[ -\frac {(1-a x)^{1-\frac {n}{2}} \left ((n-2) \left (a^2 x^2 (a x+1)^{\frac {n}{2}+1}-2^{\frac {n}{2}+1} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )\right )-2^{\frac {n}{2}+3} n \, _2F_1\left (-\frac {n}{2}-2,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )+2^{\frac {n}{2}+3} (n-1) \, _2F_1\left (-\frac {n}{2}-1,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )\right )}{4 a^4 (n-2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x^{3}\, dx \]
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 \[ \int x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )} \,d x \]
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 \[ \int x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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