3.14.57 \(\int \frac {(-2 k-(-1+k) (1+k) x+2 k x^2) (1-2 k x+k^2 x^2)}{((1-x^2) (1-k^2 x^2))^{3/4} (1-d+(1+3 d) k x-(1+3 d k^2) x^2+k (-1+d k^2) x^3)} \, dx\)

Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} k x-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} k x-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}} \]

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Rubi [F]  time = 49.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

(Sqrt[2]*k^(3/2)*Sqrt[-1 + k^2]*Sqrt[(-1 - k^2 + 2*k^2*x^2)^2]*Sqrt[(1 + k^2*(1 - 2*x^2))^2/((1 - k^2)^2*(1 -
(2*k*Sqrt[(1 - x^2)*(1 - k^2*x^2)])/(1 - k^2))^2)]*(1 - (2*k*Sqrt[(1 - x^2)*(1 - k^2*x^2)])/(1 - k^2))*Ellipti
cF[2*ArcTan[(Sqrt[2]*Sqrt[k]*((1 - x^2)*(1 - k^2*x^2))^(1/4))/Sqrt[-1 + k^2]], 1/2])/((1 - d*k^2)*(1 + k^2 - 2
*k^2*x^2)*Sqrt[(-1 - k^2*(1 - 2*x^2))^2]) + (k*(5 + (1 + 3*d)*k^2 - d*k^4)*x*((1 - x^2)/(1 - k^2*x^2))^(3/4)*(
1 - k^2*x^2)*Hypergeometric2F1[1/2, 3/4, 3/2, ((1 - k^2)*x^2)/(1 - k^2*x^2)])/((1 - d*k^2)^2*((1 - x^2)*(1 - k
^2*x^2))^(3/4)) - (k*(7 + k^2 - 3*d^2*k^2*(1 - k^2) - d*(5 + 2*k^2 + k^4))*(1 - x^2)^(3/4)*(1 - k^2*x^2)^(3/4)
*Defer[Int][1/((1 - x^2)^(3/4)*(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])/((1 - d*k^2)^2*((1 - x^2)*(1 - k^2*x^2))^(3/4)) + ((1 - k^4 - 6*d^2*k^4*(1 - k^2) - d*k^2*(19 + 14*
k^2 - k^4))*(1 - x^2)^(3/4)*(1 - k^2*x^2)^(3/4)*Defer[Int][x/((1 - x^2)^(3/4)*(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])/((1 - d*k^2)^2*((1 - x^2)*(1 - k^2*x^2))^(3/4)) + (k
*(5 + 3*(1 + 8*d)*k^2 + 3*d^2*k^4 - 3*d^2*k^6)*(1 - x^2)^(3/4)*(1 - k^2*x^2)^(3/4)*Defer[Int][x^2/((1 - x^2)^(
3/4)*(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])/((1 - d*k^2)^2*
((1 - x^2)*(1 - k^2*x^2))^(3/4))

Rubi steps

\begin {align*} \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 {(-1+k x)^2 \left (-2 k-(-1+k) (1+k) x+2 k 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 {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {(-1+k x)^2 \left (-2 k-(-1+k) (1+k) x+2 k x^2\right )}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\frac {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {(1-k x)^2 \left (-2 k+(1-k) (1+k) x+2 k x^2\right )}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\frac {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \left (\frac {k \left (5+(1+3 d) k^2-d k^4\right )}{\left (1-d k^2\right )^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}}-\frac {2 k^2 x}{\left (1-d k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}}-\frac {k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right )-\left (1-k^4-6 d^2 k^4 \left (1-k^2\right )-d \left (19 k^2+14 k^4-k^6\right )\right ) x-k \left (5+3 (1+8 d) k^2+3 d^2 k^4-3 d^2 k^6\right ) x^2}{\left (-1+d k^2\right )^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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 )}\right ) \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (2 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}} \, dx}{\left (1-d k^2\right ) \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right )-\left (1-k^4-6 d^2 k^4 \left (1-k^2\right )-d \left (19 k^2+14 k^4-k^6\right )\right ) x-k \left (5+3 (1+8 d) k^2+3 d^2 k^4-3 d^2 k^6\right ) x^2}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}+\frac {\left (k \left (5+(1+3 d) k^2-d k^4\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}} \, dx}{\left (1-d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\frac {k \left (5+(1+3 d) k^2-d k^4\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \, _2F_1\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 (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-x)^{3/4} \left (1-k^2 x\right )^{3/4}} \, dx,x,x^2\right )}{\left (1-d k^2\right ) \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \left (\frac {k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right )}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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 )}+\frac {\left (-1+k^4+6 d^2 k^4 \left (1-k^2\right )+d k^2 \left (19+14 k^2-k^4\right )\right ) x}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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 )}+\frac {k \left (-5-3 (1+8 d) k^2-3 d^2 k^4+3 d^2 k^6\right ) x^2}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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 )}\right ) \, dx}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\frac {k \left (5+(1+3 d) k^2-d k^4\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \, _2F_1\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 (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {k^2 \operatorname {Subst}\left (\int \frac {1}{\left (1+\left (-1-k^2\right ) x+k^2 x^2\right )^{3/4}} \, dx,x,x^2\right )}{1-d k^2}+\frac {\left (k \left (5+3 (1+8 d) k^2+3 d^2 k^4-3 d^2 k^6\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (\left (-1+k^4+6 d^2 k^4 \left (1-k^2\right )+d k^2 \left (19+14 k^2-k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\frac {k \left (5+(1+3 d) k^2-d k^4\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \, _2F_1\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 (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}+\frac {\left (k \left (5+3 (1+8 d) k^2+3 d^2 k^4-3 d^2 k^6\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (\left (-1+k^4+6 d^2 k^4 \left (1-k^2\right )+d k^2 \left (19+14 k^2-k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (4 k^2 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 k^2+\left (-1-k^2\right )^2+4 k^2 x^4}} \, dx,x,\sqrt [4]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}\right )}{\left (1-d k^2\right ) \left (-1-k^2+2 k^2 x^2\right )}\\ &=\frac {\sqrt {2} k^{3/2} \sqrt {-1+k^2} \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2} \sqrt {\frac {\left (1+k^2 \left (1-2 x^2\right )\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 ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {k} \sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\sqrt {-1+k^2}}\right )|\frac {1}{2}\right )}{\left (1-d k^2\right ) \left (1+k^2-2 k^2 x^2\right ) \sqrt {\left (-1-k^2 \left (1-2 x^2\right )\right )^2}}+\frac {k \left (5+(1+3 d) k^2-d k^4\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \, _2F_1\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 (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}+\frac {\left (k \left (5+3 (1+8 d) k^2+3 d^2 k^4-3 d^2 k^6\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (\left (-1+k^4+6 d^2 k^4 \left (1-k^2\right )+d k^2 \left (19+14 k^2-k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {x}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (k \left (7+k^2-3 d^2 k^2 \left (1-k^2\right )-d \left (5+2 k^2+k^4\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\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}{\left (-1+d k^2\right )^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 1.72, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

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]

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IntegrateAlgebraic [A]  time = 15.92, size = 98, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\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 {\tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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]

[Out]

ArcTan[(-d^(1/4) + d^(1/4)*k*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4) - ArcTanh[(-d^(1/4) + d^(1/4)*k*
x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="giac")

[Out]

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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 k -\left (-1+k \right ) \left (1+k \right ) x +2 k \,x^{2}\right ) \left (k^{2} x^{2}-2 k x +1\right )}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {3}{4}} \left (1-d +\left (1+3 d \right ) k x -\left (3 d \,k^{2}+1\right ) x^{2}+k \left (d \,k^{2}-1\right ) x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="maxima")

[Out]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

Timed out

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