Optimal. Leaf size=376 \[ -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 \left (4-n^2\right )}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^5 c^2 n \left (4-n^2\right )}+\frac {(1-n) (n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 (2-n)}+\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{a^5 c^2 (2-n)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{-n/2}}{a^5 c^2}+\frac {(n+3) x (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^4 c^2}-\frac {x^3 (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6150, 100, 159, 89, 79, 69, 90, 45, 37} \[ -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 \left (4-n^2\right )}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac {x^3 (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2 c^2}+\frac {(1-n) (n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 (2-n)}+\frac {(n+3) x (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^4 c^2}+\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{a^5 c^2 (2-n)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{-n/2}}{a^5 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rule 69
Rule 79
Rule 89
Rule 90
Rule 100
Rule 159
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int x^4 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{c^2}\\ &=-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {\int x^2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} (-3-a n x) \, dx}{a^2 c^2}\\ &=-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {n \int x^2 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{a^2 c^2}+\frac {(3+n) \int x^2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{a^2 c^2}\\ &=\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}+\frac {\int (1-a x)^{-n/2} (1+a x)^{-2+\frac {n}{2}} \left (-a (1-n)+a^2 n x\right ) \, dx}{a^5 c^2}+\frac {(3+n) \int (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} (-1-a n x) \, dx}{a^4 c^2}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}+\frac {n \int (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2}-\frac {\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-2-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2 (2-n)}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac {\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2 \left (4-n^2\right )}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 178, normalized size = 0.47 \[ -\frac {(1-a x)^{-\frac {n}{2}-1} \left ((a x+1)^{n/2} \left (n^2 \left (1-2 a^2 x^2\right )+6 a^2 x^2+n \left (-4 a^3 x^3+4 a^2 x^2+6 a x-4\right )+n^3 (a x-1)^2 (a x+1)-6\right )-2^{n/2} n^2 (n+2) (a x-1)^2 (a x+1) \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )\right )}{a^5 c^2 (n-2) n (n+2) (a x+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{4}}{\left (-a^{2} c \,x^{2}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________