3.16.44 \(\int \frac {((1-3 k^2) x+2 k^2 x^3) (1-2 k^2 x^2+k^4 x^4)}{((1-x^2) (1-k^2 x^2))^{3/4} (1-d+(-1+3 d k^2) x^2-3 d k^4 x^4+d k^6 x^6)} \, dx\)

Optimal. Leaf size=106 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} k^2 x^2-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} k^2 x^2-\sqrt [4]{d}}{\sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{3/4}} \]

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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]

Int[(((1 - 3*k^2)*x + 2*k^2*x^3)*(1 - 2*k^2*x^2 + k^4*x^4))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (-1 + 3*
d*k^2)*x^2 - 3*d*k^4*x^4 + d*k^6*x^6)),x]

[Out]

(-4*k^2*(1 - x^2)^(3/4)*(1 - k^2*x^2)^(3/4)*Defer[Subst][Defer[Int][(1 - k^2 + k^2*x^4)^(5/4)/(x^4 - d*(1 + k^
2*(-1 + x^4))^3), x], x, (1 - x^2)^(1/4)])/((1 - x^2)*(1 - k^2*x^2))^(3/4) + (2*(1 - 3*k^2)*(1 - x^2)^(3/4)*(1
- k^2*x^2)^(3/4)*Defer[Subst][Defer[Int][(1 - k^2 + k^2*x^4)^(5/4)/(-x^4 + d*(1 + k^2*(-1 + x^4))^3), x], x,
(1 - x^2)^(1/4)])/((1 - x^2)*(1 - k^2*x^2))^(3/4) - (4*k^2*(1 - x^2)^(3/4)*(1 - k^2*x^2)^(3/4)*Defer[Subst][De
fer[Int][(x^4*(1 - k^2 + k^2*x^4)^(5/4))/(-x^4 + d*(1 + k^2*(-1 + x^4))^3), x], x, (1 - x^2)^(1/4)])/((1 - x^2
)*(1 - k^2*x^2))^(3/4)

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*}

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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]

Integrate[(((1 - 3*k^2)*x + 2*k^2*x^3)*(1 - 2*k^2*x^2 + k^4*x^4))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (-
1 + 3*d*k^2)*x^2 - 3*d*k^4*x^4 + d*k^6*x^6)),x]

[Out]

Integrate[(((1 - 3*k^2)*x + 2*k^2*x^3)*(1 - 2*k^2*x^2 + k^4*x^4))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (-
1 + 3*d*k^2)*x^2 - 3*d*k^4*x^4 + d*k^6*x^6)), x]

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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]

IntegrateAlgebraic[(((1 - 3*k^2)*x + 2*k^2*x^3)*(1 - 2*k^2*x^2 + k^4*x^4))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1
- d + (-1 + 3*d*k^2)*x^2 - 3*d*k^4*x^4 + d*k^6*x^6)),x]

[Out]

ArcTan[(-d^(1/4) + d^(1/4)*k^2*x^2)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4) - ArcTanh[(-d^(1/4) + d^(1/4
)*k^2*x^2)/(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(((-3*k^2+1)*x+2*k^2*x^3)*(k^4*x^4-2*k^2*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(3/4)/(1-d+(3*d*k^2-1)*x^2-3*
d*k^4*x^4+d*k^6*x^6),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^{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]

integrate(((-3*k^2+1)*x+2*k^2*x^3)*(k^4*x^4-2*k^2*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(3/4)/(1-d+(3*d*k^2-1)*x^2-3*
d*k^4*x^4+d*k^6*x^6),x, algorithm="giac")

[Out]

integrate((k^4*x^4 - 2*k^2*x^2 + 1)*(2*k^2*x^3 - (3*k^2 - 1)*x)/((d*k^6*x^6 - 3*d*k^4*x^4 + (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 = 0.02, 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]

int(((-3*k^2+1)*x+2*k^2*x^3)*(k^4*x^4-2*k^2*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(3/4)/(1-d+(3*d*k^2-1)*x^2-3*d*k^4*
x^4+d*k^6*x^6),x)

[Out]

int(((-3*k^2+1)*x+2*k^2*x^3)*(k^4*x^4-2*k^2*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(3/4)/(1-d+(3*d*k^2-1)*x^2-3*d*k^4*
x^4+d*k^6*x^6),x)

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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]

integrate(((-3*k^2+1)*x+2*k^2*x^3)*(k^4*x^4-2*k^2*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(3/4)/(1-d+(3*d*k^2-1)*x^2-3*
d*k^4*x^4+d*k^6*x^6),x, algorithm="maxima")

[Out]

integrate((k^4*x^4 - 2*k^2*x^2 + 1)*(2*k^2*x^3 - (3*k^2 - 1)*x)/((d*k^6*x^6 - 3*d*k^4*x^4 + (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^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]

int(((2*k^2*x^3 - x*(3*k^2 - 1))*(k^4*x^4 - 2*k^2*x^2 + 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(3/4)*(x^2*(3*d*k^2 - 1
) - d - 3*d*k^4*x^4 + d*k^6*x^6 + 1)),x)

[Out]

int(((2*k^2*x^3 - x*(3*k^2 - 1))*(k^4*x^4 - 2*k^2*x^2 + 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(3/4)*(x^2*(3*d*k^2 - 1
) - d - 3*d*k^4*x^4 + d*k^6*x^6 + 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(((-3*k**2+1)*x+2*k**2*x**3)*(k**4*x**4-2*k**2*x**2+1)/((-x**2+1)*(-k**2*x**2+1))**(3/4)/(1-d+(3*d*k*
*2-1)*x**2-3*d*k**4*x**4+d*k**6*x**6),x)

[Out]

Timed out

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