Optimal. Leaf size=96 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} x-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} x-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}} \]
________________________________________________________________________________________
Rubi [F] time = 180.01, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1-2 x+x^2\right ) \left (-2+(-1+k) (1+k) x+2 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) x+\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx &=\int \frac {(-1+x)^2 \left (-2+(-1+k) (1+k) x+2 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) x+\left (3 d+k^2\right ) x^2+\left (-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+x)^2 \left (-2+(-1+k) (1+k) x+2 k^2 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) x+\left (3 d+k^2\right ) x^2+\left (-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-x)^2 \left (2+(1-k) (1+k) x-2 k^2 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) x-\left (3 d+k^2\right ) x^2+\left (d-k^2\right ) x^3\right )} \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 2.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-2 x+x^2\right ) \left (-2+(-1+k) (1+k) x+2 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) x+\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 15.81, size = 96, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} 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} 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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, k^{2} x^{2} + {\left (k + 1\right )} {\left (k - 1\right )} x - 2\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left ({\left (k^{2} - d\right )} x^{3} + {\left (k^{2} + 3 \, d\right )} x^{2} - {\left (3 \, d + 1\right )} x + 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]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-2 x +1\right ) \left (-2+\left (-1+k \right ) \left (1+k \right ) x +2 k^{2} x^{2}\right )}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {3}{4}} \left (-1+d -\left (1+3 d \right ) x +\left (k^{2}+3 d \right ) x^{2}+\left (k^{2}-d \right ) x^{3}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, k^{2} x^{2} + {\left (k + 1\right )} {\left (k - 1\right )} x - 2\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left ({\left (k^{2} - d\right )} x^{3} + {\left (k^{2} + 3 \, d\right )} x^{2} - {\left (3 \, d + 1\right )} x + 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]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x^2-2\,x+1\right )\,\left (2\,k^2\,x^2+x\,\left (k-1\right )\,\left (k+1\right )-2\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{3/4}\,\left (\left (d-k^2\right )\,x^3+\left (-k^2-3\,d\right )\,x^2+\left (3\,d+1\right )\,x-d+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________