\(\int x^4 (a+b \sec (c+d x^2)) \, dx\) [2]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {a x^5}{5}+b \text {Int}\left (x^4 \sec \left (c+d x^2\right ),x\right ) \]

[Out]

1/5*a*x^5+b*Unintegrable(x^4*sec(d*x^2+c),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]

[In]

Int[x^4*(a + b*Sec[c + d*x^2]),x]

[Out]

(a*x^5)/5 + b*Defer[Int][x^4*Sec[c + d*x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^4+b x^4 \sec \left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^5}{5}+b \int x^4 \sec \left (c+d x^2\right ) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]

[In]

Integrate[x^4*(a + b*Sec[c + d*x^2]),x]

[Out]

Integrate[x^4*(a + b*Sec[c + d*x^2]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int x^{4} \left (a +b \sec \left (d \,x^{2}+c \right )\right )d x\]

[In]

int(x^4*(a+b*sec(d*x^2+c)),x)

[Out]

int(x^4*(a+b*sec(d*x^2+c)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{2} + c\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*sec(d*x^2+c)),x, algorithm="fricas")

[Out]

integral(b*x^4*sec(d*x^2 + c) + a*x^4, x)

Sympy [N/A]

Not integrable

Time = 2.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int x^{4} \left (a + b \sec {\left (c + d x^{2} \right )}\right )\, dx \]

[In]

integrate(x**4*(a+b*sec(d*x**2+c)),x)

[Out]

Integral(x**4*(a + b*sec(c + d*x**2)), x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 115, normalized size of antiderivative = 7.19 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{2} + c\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*sec(d*x^2+c)),x, algorithm="maxima")

[Out]

1/5*a*x^5 + 2*b*integrate((x^4*cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) + x^4*sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + x^4
*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)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{2} + c\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*sec(d*x^2+c)),x, algorithm="giac")

[Out]

integrate((b*sec(d*x^2 + c) + a)*x^4, x)

Mupad [N/A]

Not integrable

Time = 13.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^4 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\int x^4\,\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right ) \,d x \]

[In]

int(x^4*(a + b/cos(c + d*x^2)),x)

[Out]

int(x^4*(a + b/cos(c + d*x^2)), x)