\(\int \frac {1}{x (a+b \sec (c+d x^2))^2} \, dx\) [28]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[1/(x*(a + b*Sec[c + d*x^2])^2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x \left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 6.02 (sec) , antiderivative size = 4629, normalized size of antiderivative = 257.17 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

(a^6*d*x^2*cos(2*d*x^2 + 2*c)^2*log(x) + a^6*d*x^2*log(x)*sin(2*d*x^2 + 2*c)^2 + (a^2*b^4*cos(2*c)^2 + a^2*b^4
*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2*log(x) + 4*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(c)^2 + (a^4*b^2 - 2*a^2*b^4 + b^
6)*sin(c)^2)*d*x^2*cos(d*x^2)^2*log(x) + (a^2*b^4*cos(2*c)^2 + a^2*b^4*sin(2*c)^2)*d*x^2*log(x)*sin(2*d*x^2)^2
+ 4*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(c)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6)*sin(c)^2)*d*x^2*log(x)*sin(d*x^2)^2 +
(a^6 - 2*a^4*b^2 + a^2*b^4)*d*x^2*log(x) - (a^2*b^4*sin(2*c) + 4*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^
3 - a*b^5)*sin(2*c)*sin(c))*d*x^2*cos(d*x^2)*log(x) + 2*(a^4*b^2 - a^2*b^4)*d*x^2*cos(2*c)*log(x) + 4*((a^3*b^
3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*x^2*log(x)*sin(d*x^2))*cos(2*d*x^2) - (2*a^4
*b^2*d*x^2*cos(2*d*x^2)*cos(2*c)*log(x) - 2*a^4*b^2*d*x^2*log(x)*sin(2*d*x^2)*sin(2*c) - 4*(a^5*b - a^3*b^3)*d
*x^2*cos(d*x^2)*cos(c)*log(x) + a^3*b^3*sin(d*x^2 + c) + 4*(a^5*b - a^3*b^3)*d*x^2*log(x)*sin(d*x^2)*sin(c) -
2*(a^6 - a^4*b^2)*d*x^2*log(x))*cos(2*d*x^2 + 2*c) - (a*b^5*cos(2*c)*sin(2*d*x^2) + a*b^5*cos(2*d*x^2)*sin(2*c
) - 2*(a^2*b^4 - b^6)*cos(c)*sin(d*x^2) - 2*(a^2*b^4 - b^6)*cos(d*x^2)*sin(c))*cos(d*x^2 + c) + 2*(2*(a^5*b -
2*a^3*b^3 + a*b^5)*d*x^2*cos(c)*log(x) + (a^3*b^3 - a*b^5)*sin(c))*cos(d*x^2) + (a^8*d*x^2*cos(2*d*x^2 + 2*c)^
2 + a^8*d*x^2*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + 4*((a^6*
b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^2*cos(d*x^2)^2 + 4*(a^7*b
- 2*a^5*b^3 + a^3*b^5)*d*x^2*cos(d*x^2)*cos(c) + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^2*sin(2*d*x^2)^
2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^2*sin(d*x^2)^2
- 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^2*sin(d*x^2)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d*x^2 - 2*(2*((a^5*b^
3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^2*cos(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x
^2*cos(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^2*sin(d*x^2))*
cos(2*d*x^2) - 2*(a^6*b^2*d*x^2*cos(2*d*x^2)*cos(2*c) - a^6*b^2*d*x^2*sin(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b
^3)*d*x^2*cos(d*x^2)*cos(c) + 2*(a^7*b - a^5*b^3)*d*x^2*sin(d*x^2)*sin(c) - (a^8 - a^6*b^2)*d*x^2)*cos(2*d*x^2
+ 2*c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^2*cos(d*x^2) -
2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^2*sin(d*x^2) + (a^6*b^2 - a^
4*b^4)*d*x^2*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^2*cos(2*c)*sin(2*d*x^2) + a^6*b^2*d*x^2*cos(2*d*x^2)*sin(
2*c) - 2*(a^7*b - a^5*b^3)*d*x^2*cos(c)*sin(d*x^2) - 2*(a^7*b - a^5*b^3)*d*x^2*cos(d*x^2)*sin(c))*sin(2*d*x^2
+ 2*c))*integrate(-2*(a^2*b^4*cos(2*c)*sin(2*d*x^2) + a^2*b^4*cos(2*d*x^2)*sin(2*c) - 2*(a^3*b^3 - a*b^5)*cos(
c)*sin(d*x^2) - 2*(a^3*b^3 - a*b^5)*cos(d*x^2)*sin(c) + (a^3*b^3*sin(d*x^2 + c) + (2*a^5*b - a^3*b^3)*d*x^2*co
s(d*x^2 + c))*cos(2*d*x^2 + 2*c) + ((2*a^5*b - 3*a^3*b^3 + a*b^5)*d*x^2 + (a*b^5*sin(2*c) - (2*a^3*b^3 - a*b^5
)*d*x^2*cos(2*c))*cos(2*d*x^2) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*cos(c) - (a^2*b^4 - b^6)*sin(c))*cos(d
*x^2) + (a*b^5*cos(2*c) + (2*a^3*b^3 - a*b^5)*d*x^2*sin(2*c))*sin(2*d*x^2) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*
d*x^2*sin(c) + (a^2*b^4 - b^6)*cos(c))*sin(d*x^2))*cos(d*x^2 + c) - (a^3*b^3*cos(d*x^2 + c) + a^4*b^2 - (2*a^5
*b - a^3*b^3)*d*x^2*sin(d*x^2 + c))*sin(2*d*x^2 + 2*c) + (a^3*b^3 - a*b^5 - (a*b^5*cos(2*c) + (2*a^3*b^3 - a*b
^5)*d*x^2*sin(2*c))*cos(2*d*x^2) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*sin(c) + (a^2*b^4 - b^6)*cos(c))*cos
(d*x^2) + (a*b^5*sin(2*c) - (2*a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*sin(2*d*x^2) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6
)*d*x^2*cos(c) - (a^2*b^4 - b^6)*sin(c))*sin(d*x^2))*sin(d*x^2 + c))/(a^8*d*x^3*cos(2*d*x^2 + 2*c)^2 + a^8*d*x
^3*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^3*cos(2*d*x^2)^2 + 4*((a^6*b^2 - 2*a^4
*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^3*cos(d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3
+ a^3*b^5)*d*x^3*cos(d*x^2)*cos(c) + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 + 4*((a^6
*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^3*sin(d*x^2)^2 - 4*(a^7*b
- 2*a^5*b^3 + a^3*b^5)*d*x^3*sin(d*x^2)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d*x^3 - 2*(2*((a^5*b^3 - a^3*b^5
)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^3*cos(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^3*cos(2*c)
+ 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^3*sin(d*x^2))*cos(2*d*x^2
) - 2*(a^6*b^2*d*x^3*cos(2*d*x^2)*cos(2*c) - a^6*b^2*d*x^3*sin(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^3*c
os(d*x^2)*cos(c) + 2*(a^7*b - a^5*b^3)*d*x^3*sin(d*x^2)*sin(c) - (a^8 - a^6*b^2)*d*x^3)*cos(2*d*x^2 + 2*c) + 2
*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^3*cos(d*x^2) - 2*((a^5*b^3
- a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^3*sin(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^
3*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^3*cos(2*c)*sin(2*d*x^2) + a^6*b^2*d*x^3*cos(2*d*x^2)*sin(2*c) - 2*(a
^7*b - a^5*b^3)*d*x^3*cos(c)*sin(d*x^2) - 2*(a^7*b - a^5*b^3)*d*x^3*cos(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c)), x)
- (a^2*b^4*cos(2*c) - 4*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*x^2*cos(d*x
^2)*log(x) + 4*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*d*x^2*log(x)*sin(d*x^2)
- 2*(a^4*b^2 - a^2*b^4)*d*x^2*log(x)*sin(2*c))*sin(2*d*x^2) - (2*a^4*b^2*d*x^2*cos(2*c)*log(x)*sin(2*d*x^2) +
2*a^4*b^2*d*x^2*cos(2*d*x^2)*log(x)*sin(2*c) - a^3*b^3*cos(d*x^2 + c) - 4*(a^5*b - a^3*b^3)*d*x^2*cos(c)*log(
x)*sin(d*x^2) - 4*(a^5*b - a^3*b^3)*d*x^2*cos(d*x^2)*log(x)*sin(c) - a^4*b^2)*sin(2*d*x^2 + 2*c) + (a*b^5*cos(
2*d*x^2)*cos(2*c) - a*b^5*sin(2*d*x^2)*sin(2*c) - a^3*b^3 + a*b^5 - 2*(a^2*b^4 - b^6)*cos(d*x^2)*cos(c) + 2*(a
^2*b^4 - b^6)*sin(d*x^2)*sin(c))*sin(d*x^2 + c) - 2*(2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*log(x)*sin(c) - (a^3*
b^3 - a*b^5)*cos(c))*sin(d*x^2))/(a^8*d*x^2*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^2*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*c
os(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2
- 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^2*cos(d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^2*cos(d*x^2)*cos(c)
+ (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^2*sin(2*d*x^2)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2
+ (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^2*sin(d*x^2)^2 - 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^2*sin(d*
x^2)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d*x^2 - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b
^5)*sin(2*c)*sin(c))*d*x^2*cos(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^2*cos(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin
(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^2*sin(d*x^2))*cos(2*d*x^2) - 2*(a^6*b^2*d*x^2*cos(2*d*x^2)*co
s(2*c) - a^6*b^2*d*x^2*sin(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^2*cos(d*x^2)*cos(c) + 2*(a^7*b - a^5*b^
3)*d*x^2*sin(d*x^2)*sin(c) - (a^8 - a^6*b^2)*d*x^2)*cos(2*d*x^2 + 2*c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(
2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^2*cos(d*x^2) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b
^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^2*sin(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^2*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^
2*d*x^2*cos(2*c)*sin(2*d*x^2) + a^6*b^2*d*x^2*cos(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^2*cos(c)*sin(d*x
^2) - 2*(a^7*b - a^5*b^3)*d*x^2*cos(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c))

Giac [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x\,{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

int(1/(x*(a + b/cos(c + d*x^2))^2),x)

[Out]

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