Optimal. Leaf size=68 \[ -\frac {c 2^{\frac {n}{2}+2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-1,2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6140, 69} \[ -\frac {c 2^{\frac {n}{2}+2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-1,2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 6140
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^{1-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}} \, dx\\ &=-\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 65, normalized size = 0.96 \[ \frac {c 2^{\frac {n}{2}+2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-1,2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-4)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a^{2} c x^{2} - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (a^{2} c x^{2} - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-a^2\,c\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - c \left (\int a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________