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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0237837, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 7.30275, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

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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Maple [A]  time = 0.369, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b\sec \left ( d{x}^{2}+c \right ) \right ) ^{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a + b \sec{\left (c + d x^{2} \right )}\right )^{2}}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x^{3}}\,{d x} \end{align*}

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)