3.4 \(\int x^2 (a+b \sec (c+d x^2)) \, dx\)

Optimal. Leaf size=25 \[ b \text{Unintegrable}\left (x^2 \sec \left (c+d x^2\right ),x\right )+\frac{a x^3}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0152668, 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 = {} \[ \int x^2 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*Sec[c + d*x^2]),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.727728, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\sec \left ( d{x}^{2}+c \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{2} \sec \left (d x^{2} + c\right ) + a x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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