Integrand size = 95, antiderivative size = 98 \[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\frac {\arctan \left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} k x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} k x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}} \]
arctan((-d^(1/4)+d^(1/4)*k*x)/(1+(-k^2-1)*x^2+k^2*x^4)^(1/4))/d^(3/4)-arct anh((-d^(1/4)+d^(1/4)*k*x)/(1+(-k^2-1)*x^2+k^2*x^4)^(1/4))/d^(3/4)
\[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx \]
Integrate[((-2*k - (-1 + k)*(1 + k)*x + 2*k*x^2)*(1 - 2*k*x + k^2*x^2))/(( (1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (1 + 3*d)*k*x - (1 + 3*d*k^2)*x^2 + k*(-1 + d*k^2)*x^3)),x]
Integrate[((-2*k - (-1 + k)*(1 + k)*x + 2*k*x^2)*(1 - 2*k*x + k^2*x^2))/(( (1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (1 + 3*d)*k*x - (1 + 3*d*k^2)*x^2 + k*(-1 + d*k^2)*x^3)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (2 k x^2-(k-1) (k+1) x-2 k\right ) \left (k^2 x^2-2 k x+1\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (k x^3 \left (d k^2-1\right )-x^2 \left (3 d k^2+1\right )+(3 d+1) k x-d+1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {\left (2 k x^2-(k-1) (k+1) x-2 k\right ) \left (k^2 x^2-2 k x+1\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (k x^3 \left (d k^2-1\right )-x^2 \left (3 d k^2+1\right )+(3 d+1) k x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-2 k x^2-(1-k) (k+1) x+2 k\right ) \left (-k^2 x^2+2 k x-1\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (-k x^3 \left (1-d k^2\right )-x^2 \left (3 d k^2+1\right )+(3 d+1) k x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {k \left (-3 d^2 \left (1-k^2\right ) k^2-d \left (k^4+2 k^2+5\right )+k^2+7\right )-k x^2 \left (-3 d^2 k^6+3 d^2 k^4+3 (8 d+1) k^2+5\right )-x \left (-6 d^2 \left (1-k^2\right ) k^4-d \left (-k^6+14 k^4+19 k^2\right )-k^4+1\right )}{\left (d k^2-1\right )^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (-k x^3 \left (1-d k^2\right )-x^2 \left (3 d k^2+1\right )+(3 d+1) k x-d+1\right )}-\frac {2 k^2 x}{\left (1-d k^2\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}+\frac {k \left (-d k^4+(3 d+1) k^2+5\right )}{\left (1-d k^2\right )^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {k \left (-3 d^2 \left (1-k^2\right ) k^2-d \left (k^4+2 k^2+5\right )+k^2+7\right ) \int \frac {1}{\left (-k \left (1-d k^2\right ) x^3-\left (3 d k^2+1\right ) x^2+(3 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (1-d k^2\right )^2}+\frac {\left (-6 d^2 \left (1-k^2\right ) k^4-d \left (-k^4+14 k^2+19\right ) k^2-k^4+1\right ) \int \frac {x}{\left (-k \left (1-d k^2\right ) x^3-\left (3 d k^2+1\right ) x^2+(3 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (1-d k^2\right )^2}+\frac {k \left (-3 d^2 k^6+3 d^2 k^4+3 (8 d+1) k^2+5\right ) \int \frac {x^2}{\left (-k \left (1-d k^2\right ) x^3-\left (3 d k^2+1\right ) x^2+(3 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (1-d k^2\right )^2}+\frac {\sqrt {2} \sqrt {k^2-1} k^{3/2} \sqrt {\left (2 k^2 x^2-k^2-1\right )^2} \sqrt {\frac {\left (k^2 \left (1-2 x^2\right )+1\right )^2}{\left (1-k^2\right )^2 \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right )^2}} \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt {k} \sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\sqrt {k^2-1}}\right ),\frac {1}{2}\right )}{\left (1-d k^2\right ) \left (-2 k^2 x^2+k^2+1\right ) \sqrt {\left (-\left (k^2 \left (1-2 x^2\right )\right )-1\right )^2}}+\frac {k x \left (-d k^4+(3 d+1) k^2+5\right ) \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {\left (1-k^2\right ) x^2}{1-k^2 x^2}\right )}{\left (1-d k^2\right )^2 \left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{3/4}}\) |
Int[((-2*k - (-1 + k)*(1 + k)*x + 2*k*x^2)*(1 - 2*k*x + k^2*x^2))/(((1 - x ^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (1 + 3*d)*k*x - (1 + 3*d*k^2)*x^2 + k*(- 1 + d*k^2)*x^3)),x]
3.14.57.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
\[\int \frac {\left (-2 k -\left (-1+k \right ) \left (1+k \right ) x +2 k \,x^{2}\right ) \left (k^{2} x^{2}-2 k x +1\right )}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {3}{4}} \left (1-d +\left (1+3 d \right ) k x -\left (3 d \,k^{2}+1\right ) x^{2}+k \left (d \,k^{2}-1\right ) x^{3}\right )}d x\]
int((-2*k-(-1+k)*(1+k)*x+2*k*x^2)*(k^2*x^2-2*k*x+1)/((-x^2+1)*(-k^2*x^2+1) )^(3/4)/(1-d+(1+3*d)*k*x-(3*d*k^2+1)*x^2+k*(d*k^2-1)*x^3),x)
int((-2*k-(-1+k)*(1+k)*x+2*k*x^2)*(k^2*x^2-2*k*x+1)/((-x^2+1)*(-k^2*x^2+1) )^(3/4)/(1-d+(1+3*d)*k*x-(3*d*k^2+1)*x^2+k*(d*k^2-1)*x^3),x)
Timed out. \[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\text {Timed out} \]
integrate((-2*k-(-1+k)*(1+k)*x+2*k*x^2)*(k^2*x^2-2*k*x+1)/((-x^2+1)*(-k^2* x^2+1))^(3/4)/(1-d+(1+3*d)*k*x-(3*d*k^2+1)*x^2+k*(d*k^2-1)*x^3),x, algorit hm="fricas")
Timed out. \[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\text {Timed out} \]
integrate((-2*k-(-1+k)*(1+k)*x+2*k*x**2)*(k**2*x**2-2*k*x+1)/((-x**2+1)*(- k**2*x**2+1))**(3/4)/(1-d+(1+3*d)*k*x-(3*d*k**2+1)*x**2+k*(d*k**2-1)*x**3) ,x)
\[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\int { -\frac {{\left (k^{2} x^{2} - 2 \, k x + 1\right )} {\left ({\left (k + 1\right )} {\left (k - 1\right )} x - 2 \, k x^{2} + 2 \, k\right )}}{{\left ({\left (d k^{2} - 1\right )} k x^{3} + {\left (3 \, d + 1\right )} k x - {\left (3 \, d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {3}{4}}} \,d x } \]
integrate((-2*k-(-1+k)*(1+k)*x+2*k*x^2)*(k^2*x^2-2*k*x+1)/((-x^2+1)*(-k^2* x^2+1))^(3/4)/(1-d+(1+3*d)*k*x-(3*d*k^2+1)*x^2+k*(d*k^2-1)*x^3),x, algorit hm="maxima")
-integrate((k^2*x^2 - 2*k*x + 1)*((k + 1)*(k - 1)*x - 2*k*x^2 + 2*k)/(((d* k^2 - 1)*k*x^3 + (3*d + 1)*k*x - (3*d*k^2 + 1)*x^2 - d + 1)*((k^2*x^2 - 1) *(x^2 - 1))^(3/4)), x)
\[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\int { -\frac {{\left (k^{2} x^{2} - 2 \, k x + 1\right )} {\left ({\left (k + 1\right )} {\left (k - 1\right )} x - 2 \, k x^{2} + 2 \, k\right )}}{{\left ({\left (d k^{2} - 1\right )} k x^{3} + {\left (3 \, d + 1\right )} k x - {\left (3 \, d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {3}{4}}} \,d x } \]
integrate((-2*k-(-1+k)*(1+k)*x+2*k*x^2)*(k^2*x^2-2*k*x+1)/((-x^2+1)*(-k^2* x^2+1))^(3/4)/(1-d+(1+3*d)*k*x-(3*d*k^2+1)*x^2+k*(d*k^2-1)*x^3),x, algorit hm="giac")
integrate(-(k^2*x^2 - 2*k*x + 1)*((k + 1)*(k - 1)*x - 2*k*x^2 + 2*k)/(((d* k^2 - 1)*k*x^3 + (3*d + 1)*k*x - (3*d*k^2 + 1)*x^2 - d + 1)*((k^2*x^2 - 1) *(x^2 - 1))^(3/4)), x)
Timed out. \[ \int \frac {\left (-2 k-(-1+k) (1+k) x+2 k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+(1+3 d) k x-\left (1+3 d k^2\right ) x^2+k \left (-1+d k^2\right ) x^3\right )} \, dx=\int -\frac {\left (-2\,k\,x^2+\left (k-1\right )\,\left (k+1\right )\,x+2\,k\right )\,\left (k^2\,x^2-2\,k\,x+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{3/4}\,\left (k\,\left (d\,k^2-1\right )\,x^3+\left (-3\,d\,k^2-1\right )\,x^2+k\,\left (3\,d+1\right )\,x-d+1\right )} \,d x \]
int(-((2*k - 2*k*x^2 + x*(k - 1)*(k + 1))*(k^2*x^2 - 2*k*x + 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(3/4)*(k*x*(3*d + 1) - x^2*(3*d*k^2 + 1) - d + k*x^3*(d* k^2 - 1) + 1)),x)