\(\int \frac {1}{x (a+b \tan (c+d x^2))^2} \, dx\) [23]

Optimal result

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

[Out]

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

[In]

Int[1/(x*(a + b*Tan[c + d*x^2])^2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.32 (sec) , antiderivative size = 3616, normalized size of antiderivative = 200.89 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

(((4*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 - 3*a^2*b^8 - b^10)*cos(2*c)^2 + (4*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 - 3*a
^2*b^8 - b^10)*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2*log(x) + (a^10 - a^8*b^2)*d*x^2*cos(2*d*x^2 + 2*c)^2*log(x) +
((4*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 - 3*a^2*b^8 - b^10)*cos(2*c)^2 + (4*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 - 3*a^
2*b^8 - b^10)*sin(2*c)^2)*d*x^2*log(x)*sin(2*d*x^2)^2 + (a^10 - a^8*b^2)*d*x^2*log(x)*sin(2*d*x^2 + 2*c)^2 + (
a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*d*x^2*log(x) - (2*((a^6*b^4 + a^4*b^6 - a^2*b^8 -
b^10)*cos(2*c) - 2*(a^9*b + 2*a^7*b^3 - 2*a^3*b^7 - a*b^9)*sin(2*c))*d*x^2*log(x) + 2*(a^7*b^3 + 3*a^5*b^5 +
3*a^3*b^7 + a*b^9)*cos(2*c) + (a^4*b^6 + 2*a^2*b^8 + b^10)*sin(2*c))*cos(2*d*x^2) - 2*(((a^6*b^4 - a^4*b^6)*co
s(2*c) - 2*(a^9*b - a^5*b^5)*sin(2*c))*d*x^2*cos(2*d*x^2)*log(x) - (2*(a^9*b - a^5*b^5)*cos(2*c) + (a^6*b^4 -
a^4*b^6)*sin(2*c))*d*x^2*log(x)*sin(2*d*x^2) - (a^10 + a^8*b^2 - a^6*b^4 - a^4*b^6)*d*x^2*log(x))*cos(2*d*x^2
+ 2*c) - (((4*a^10*b^2 + 16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*cos(2*c)^2 + (4*a^10*b^2 +
16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + (a^12 + 2*a^10*b^
2 + a^8*b^4)*d*x^2*cos(2*d*x^2 + 2*c)^2 + ((4*a^10*b^2 + 16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b
^12)*cos(2*c)^2 + (4*a^10*b^2 + 16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*sin(2*c)^2)*d*x^2*si
n(2*d*x^2)^2 + (a^12 + 2*a^10*b^2 + a^8*b^4)*d*x^2*sin(2*d*x^2 + 2*c)^2 - 2*((a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8
+ 4*a^2*b^10 + b^12)*cos(2*c) - 2*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*sin(2*c)
)*d*x^2*cos(2*d*x^2) + 2*(2*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*cos(2*c) + (a^
8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*sin(2*c))*d*x^2*sin(2*d*x^2) + (a^12 + 6*a^10*b^2 + 15*a^8*
b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*x^2 - 2*(((a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*cos(2*c) - 2*(a
^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*sin(2*c))*d*x^2*cos(2*d*x^2) - (2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a
^5*b^7)*cos(2*c) + (a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*sin(2*c))*d*x^2*sin(2*d*x^2) - (a^12 + 4*a^10*b^2 + 6*a^8*b
^4 + 4*a^6*b^6 + a^4*b^8)*d*x^2)*cos(2*d*x^2 + 2*c) - 2*((2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*cos(2*c
) + (a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*sin(2*c))*d*x^2*cos(2*d*x^2) + ((a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*cos(2*c) -
2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*sin(2*c))*d*x^2*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))*integrate(2*((
b^6*sin(2*c) - 2*(a*b^5*sin(2*c) + 2*(a^4*b^2 + a^2*b^4)*cos(2*c))*d*x^2 + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2
*d*x^2) + (b^6*cos(2*c) - 2*(a*b^5*cos(2*c) - 2*(a^4*b^2 + a^2*b^4)*sin(2*c))*d*x^2 - 2*(a^3*b^3 + a*b^5)*sin(
2*c))*sin(2*d*x^2) + (2*a^5*b*d*x^2 - a^4*b^2)*sin(2*d*x^2 + 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 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2
*b^6 + b^8)*sin(2*c)^2)*d*x^3*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*
b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8)*cos(2*c)
- 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(2*c))*d*x^3*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5
+ a*b^7)*cos(2*c) + (a^4*b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x^3*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d*x^3 - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^3*cos(2*d*x^2) - (a^4*b^4*s
in(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^3)*cos(2*d*x^2 +
2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b
+ a^5*b^3)*sin(2*c))*d*x^3*sin(2*d*x^2))*sin(2*d*x^2 + 2*c)), x) + (2*(2*(a^9*b + 2*a^7*b^3 - 2*a^3*b^7 - a*b^
9)*cos(2*c) + (a^6*b^4 + a^4*b^6 - a^2*b^8 - b^10)*sin(2*c))*d*x^2*log(x) - (a^4*b^6 + 2*a^2*b^8 + b^10)*cos(2
*c) + 2*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*sin(2*c))*sin(2*d*x^2) + (a^8*b^2 + 2*a^6*b^4 + a^4*b^6 - 2*
(2*(a^9*b - a^5*b^5)*cos(2*c) + (a^6*b^4 - a^4*b^6)*sin(2*c))*d*x^2*cos(2*d*x^2)*log(x) - 2*((a^6*b^4 - a^4*b^
6)*cos(2*c) - 2*(a^9*b - a^5*b^5)*sin(2*c))*d*x^2*log(x)*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))/(((4*a^10*b^2 + 16*
a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*cos(2*c)^2 + (4*a^10*b^2 + 16*a^8*b^4 + 24*a^6*b^6 + 17
*a^4*b^8 + 6*a^2*b^10 + b^12)*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + (a^12 + 2*a^10*b^2 + a^8*b^4)*d*x^2*cos(2*d*x
^2 + 2*c)^2 + ((4*a^10*b^2 + 16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*cos(2*c)^2 + (4*a^10*b^
2 + 16*a^8*b^4 + 24*a^6*b^6 + 17*a^4*b^8 + 6*a^2*b^10 + b^12)*sin(2*c)^2)*d*x^2*sin(2*d*x^2)^2 + (a^12 + 2*a^1
0*b^2 + a^8*b^4)*d*x^2*sin(2*d*x^2 + 2*c)^2 - 2*((a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*cos(2*c
) - 2*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*sin(2*c))*d*x^2*cos(2*d*x^2) + 2*(2*
(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*cos(2*c) + (a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^
8 + 4*a^2*b^10 + b^12)*sin(2*c))*d*x^2*sin(2*d*x^2) + (a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^
8 + 6*a^2*b^10 + b^12)*d*x^2 - 2*(((a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*cos(2*c) - 2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^
5 + a^5*b^7)*sin(2*c))*d*x^2*cos(2*d*x^2) - (2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*cos(2*c) + (a^8*b^4
+ 2*a^6*b^6 + a^4*b^8)*sin(2*c))*d*x^2*sin(2*d*x^2) - (a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8)*d*
x^2)*cos(2*d*x^2 + 2*c) - 2*((2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*cos(2*c) + (a^8*b^4 + 2*a^6*b^6 + a
^4*b^8)*sin(2*c))*d*x^2*cos(2*d*x^2) + ((a^8*b^4 + 2*a^6*b^6 + a^4*b^8)*cos(2*c) - 2*(a^11*b + 3*a^9*b^3 + 3*a
^7*b^5 + a^5*b^7)*sin(2*c))*d*x^2*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))

Giac [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2} \,d x \]

[In]

int(1/(x*(a + b*tan(c + d*x^2))^2),x)

[Out]

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