Optimal. Leaf size=106 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} k^2 x^2-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} k^2 x^2-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}} \]
________________________________________________________________________________________
Rubi [F] time = 12.19, 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 (\left (1-3 k^2\right ) x+2 k^2 x^3\right ) \left (1-2 k^2 x^2+k^4 x^4\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x^2-3 d k^4 x^4+d k^6 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (\left (1-3 k^2\right ) x+2 k^2 x^3\right ) \left (1-2 k^2 x^2+k^4 x^4\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x^2-3 d k^4 x^4+d k^6 x^6\right )} \, dx &=\frac {\int \frac {\left (-k^2+k^4 x^2\right )^2 \left (\left (1-3 k^2\right ) x+2 k^2 x^3\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x^2-3 d k^4 x^4+d k^6 x^6\right )} \, dx}{k^4}\\ &=\frac {\int \frac {x \left (1-3 k^2+2 k^2 x^2\right ) \left (-k^2+k^4 x^2\right )^2}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x^2-3 d k^4 x^4+d k^6 x^6\right )} \, dx}{k^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-3 k^2+2 k^2 x\right ) \left (-k^2+k^4 x\right )^2}{\left ((1-x) \left (1-k^2 x\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )} \, dx,x,x^2\right )}{2 k^4}\\ &=\frac {\left (\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-3 k^2+2 k^2 x\right ) \left (-k^2+k^4 x\right )^2}{(1-x)^{3/4} \left (1-k^2 x\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )} \, dx,x,x^2\right )}{2 k^4 \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 ) \operatorname {Subst}\left (\int \frac {\left (1-k^2 x\right )^{5/4} \left (1-3 k^2+2 k^2 x\right )}{(1-x)^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )} \, dx,x,x^2\right )}{2 \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 ) \operatorname {Subst}\left (\int \frac {\left (1-k^2 x\right )^{5/4} \left (1-3 k^2+2 k^2 x\right )}{(1-x)^{3/4} \left (1-d-\left (1-3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )} \, dx,x,x^2\right )}{2 \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 ) \operatorname {Subst}\left (\int \left (\frac {3 \left (1-\frac {1}{3 k^2}\right ) k^2 \left (1-k^2 x\right )^{5/4}}{(1-x)^{3/4} \left (-1+d+\left (1-3 d k^2\right ) x+3 d k^4 x^2-d k^6 x^3\right )}+\frac {2 k^2 x \left (1-k^2 x\right )^{5/4}}{(1-x)^{3/4} \left (1-d-\left (1-3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )}\right ) \, dx,x,x^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 {x \left (1-k^2 x\right )^{5/4}}{(1-x)^{3/4} \left (1-d-\left (1-3 d k^2\right ) x-3 d k^4 x^2+d k^6 x^3\right )} \, dx,x,x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}+\frac {\left (\left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2 x\right )^{5/4}}{(1-x)^{3/4} \left (-1+d+\left (1-3 d k^2\right ) x+3 d k^4 x^2-d k^6 x^3\right )} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right ) \left (1+k^2 \left (-1+x^4\right )\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (2 \left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1+k^2 \left (-1+x^4\right )\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right ) \left (1-k^2+k^2 x^4\right )^{5/4}}{x^4-d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (2 \left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-d \left (1-k^2 \left (3-3 k^2+k^4\right )\right )+\left (1-3 d k^2 \left (-1+k^2\right )^2\right ) x^4-3 d k^4 \left (1-k^2\right ) x^8-d k^6 x^{12}}+\frac {x^4 \left (1-k^2+k^2 x^4\right )^{5/4}}{d \left (1-k^2 \left (3-3 k^2+k^4\right )\right )-\left (1-3 d k^2 \left (-1+k^2\right )^2\right ) x^4+3 d k^4 \left (1-k^2\right ) x^8+d k^6 x^{12}}\right ) \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (2 \left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-d \left (1-k^2 \left (3-3 k^2+k^4\right )\right )+\left (1-3 d k^2 \left (-1+k^2\right )^2\right ) x^4-3 d k^4 \left (1-k^2\right ) x^8-d k^6 x^{12}} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-k^2+k^2 x^4\right )^{5/4}}{d \left (1-k^2 \left (3-3 k^2+k^4\right )\right )-\left (1-3 d k^2 \left (-1+k^2\right )^2\right ) x^4+3 d k^4 \left (1-k^2\right ) x^8+d k^6 x^{12}} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (2 \left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ &=-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{x^4-d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (4 k^2 \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-k^2+k^2 x^4\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}-\frac {\left (2 \left (-1+3 k^2\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (1-k^2+k^2 x^4\right )^{5/4}}{-x^4+d \left (1+k^2 \left (-1+x^4\right )\right )^3} \, dx,x,\sqrt [4]{1-x^2}\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (1-3 k^2\right ) x+2 k^2 x^3\right ) \left (1-2 k^2 x^2+k^4 x^4\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d+\left (-1+3 d k^2\right ) x^2-3 d k^4 x^4+d k^6 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 15.91, size = 106, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} k^2 x^2}{\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^2 x^2}{\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 (k^{4} x^{4} - 2 \, k^{2} x^{2} + 1\right )} {\left (2 \, k^{2} x^{3} - {\left (3 \, k^{2} - 1\right )} x\right )}}{{\left (d k^{6} x^{6} - 3 \, d k^{4} x^{4} + {\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]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-3 k^{2}+1\right ) x +2 k^{2} x^{3}\right ) \left (k^{4} x^{4}-2 k^{2} x^{2}+1\right )}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {3}{4}} \left (1-d +\left (3 d \,k^{2}-1\right ) x^{2}-3 d \,k^{4} x^{4}+d \,k^{6} x^{6}\right )}\, dx\]
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 (k^{4} x^{4} - 2 \, k^{2} x^{2} + 1\right )} {\left (2 \, k^{2} x^{3} - {\left (3 \, k^{2} - 1\right )} x\right )}}{{\left (d k^{6} x^{6} - 3 \, d k^{4} x^{4} + {\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]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,k^2\,x^3-x\,\left (3\,k^2-1\right )\right )\,\left (k^4\,x^4-2\,k^2\,x^2+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{3/4}\,\left (x^2\,\left (3\,d\,k^2-1\right )-d-3\,d\,k^4\,x^4+d\,k^6\,x^6+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]
________________________________________________________________________________________