3.26 \(\int \frac {x^2}{(a+b \sec (c+d x^2))^2} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^2/(a+b*sec(d*x^2+c))^2,x)

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2/(a + b*Sec[c + d*x^2])^2,x]

[Out]

Defer[Int][x^2/(a + b*Sec[c + d*x^2])^2, x]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx &=\int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 5.80, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2/(a + b*Sec[c + d*x^2])^2,x]

[Out]

Integrate[x^2/(a + b*Sec[c + d*x^2])^2, x]

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{b^{2} \sec \left (d x^{2} + c\right )^{2} + 2 \, a b \sec \left (d x^{2} + c\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*sec(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(x^2/(b^2*sec(d*x^2 + c)^2 + 2*a*b*sec(d*x^2 + c) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*sec(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^2/(b*sec(d*x^2 + c) + a)^2, x)

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maple [A]  time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +b \sec \left (d \,x^{2}+c \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*sec(d*x^2+c))^2,x)

[Out]

int(x^2/(a+b*sec(d*x^2+c))^2,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*sec(d*x^2+c))^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x^2}{{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/cos(c + d*x^2))^2,x)

[Out]

int(x^2/(a + b/cos(c + d*x^2))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*sec(d*x**2+c))**2,x)

[Out]

Integral(x**2/(a + b*sec(c + d*x**2))**2, x)

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