3.765 \(\int \frac {\text {F0}(x)}{(x^2+\text {F0}(x))^2} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{\text {F0}(x)+x^2},x\right )-\text {Int}\left (\frac {x^2}{\left (\text {F0}(x)+x^2\right )^2},x\right ) \]

[Out]

-CannotIntegrate(x^2/(x^2+F0(x))^2,x)+CannotIntegrate(1/(x^2+F0(x)),x)

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\text {F0}(x)}{\left (x^2+\text {F0}(x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[F0[x]/(x^2 + F0[x])^2,x]

[Out]

-Defer[Int][x^2/(x^2 + F0[x])^2, x] + Defer[Int][(x^2 + F0[x])^(-1), x]

Rubi steps

\begin {align*} \int \frac {\text {F0}(x)}{\left (x^2+\text {F0}(x)\right )^2} \, dx &=\int \left (-\frac {x^2}{\left (x^2+\text {F0}(x)\right )^2}+\frac {1}{x^2+\text {F0}(x)}\right ) \, dx\\ &=-\int \frac {x^2}{\left (x^2+\text {F0}(x)\right )^2} \, dx+\int \frac {1}{x^2+\text {F0}(x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\text {F0}(x)}{\left (x^2+\text {F0}(x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[F0[x]/(x^2 + F0[x])^2,x]

[Out]

Integrate[F0[x]/(x^2 + F0[x])^2, x]

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fricas [A]  time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F_{0}\relax (x)}{x^{4} + 2 \, x^{2} F_{0}\relax (x) + F_{0}\relax (x)^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x^2+F0(x))^2,x, algorithm="fricas")

[Out]

integral(F0(x)/(x^4 + 2*x^2*F0(x) + F0(x)^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F_{0}\relax (x)}{{\left (x^{2} + F_{0}\relax (x)\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x^2+F0(x))^2,x, algorithm="giac")

[Out]

integrate(F0(x)/(x^2 + F0(x))^2, x)

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\mathit {F0} \relax (x )}{\left (x^{2}+\mathit {F0} \relax (x )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F0(x)/(x^2+F0(x))^2,x)

[Out]

int(F0(x)/(x^2+F0(x))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F_{0}\relax (x)}{{\left (x^{2} + F_{0}\relax (x)\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x^2+F0(x))^2,x, algorithm="maxima")

[Out]

integrate(F0(x)/(x^2 + F0(x))^2, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {F_{0}\relax (x)}{{\left (F_{0}\relax (x)+x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F0(x)/(F0(x) + x^2)^2,x)

[Out]

int(F0(x)/(F0(x) + x^2)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {F_{0}}{\relax (x )}}{\left (x^{2} + \operatorname {F_{0}}{\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x**2+F0(x))**2,x)

[Out]

Integral(F0(x)/(x**2 + F0(x))**2, x)

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