3.30 \(\int \frac {1}{x^3 (a+b \sec (c+d x^2))^2} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^3/(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 = {} \[ \int \frac {1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^3*(a + b*Sec[c + d*x^2])^2),x]

[Out]

Defer[Int][1/(x^3*(a + b*Sec[c + d*x^2])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx &=\int \frac {1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 7.81, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^3*(a + b*Sec[c + d*x^2])^2),x]

[Out]

Integrate[1/(x^3*(a + b*Sec[c + d*x^2])^2), x]

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fricas [A]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{3} \sec \left (d x^{2} + c\right )^{2} + 2 \, a b x^{3} \sec \left (d x^{2} + c\right ) + a^{2} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*sec(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^3*sec(d*x^2 + c)^2 + 2*a*b*x^3*sec(d*x^2 + c) + a^2*x^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*sec(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*sec(d*x^2 + c) + a)^2*x^3), x)

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maple [A]  time = 1.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a +b \sec \left (d \,x^{2}+c \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b*sec(d*x^2+c))^2,x)

[Out]

int(1/x^3/(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(1/x^3/(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 \[ \int \frac {1}{x^3\,{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/cos(c + d*x^2))^2),x)

[Out]

int(1/(x^3*(a + b/cos(c + d*x^2))^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*sec(d*x**2+c))**2,x)

[Out]

Integral(1/(x**3*(a + b*sec(c + d*x**2))**2), x)

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