\(\int \frac {1}{x (a+b \tan (c+d \sqrt {x}))^2} \, dx\) [45]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 168.73 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.51 (sec) , antiderivative size = 3514, normalized size of antiderivative = 175.70 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

((((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*cos(2*d*sqrt(x))^2 + (a^12 + 2*a^10*b^2 + a^8*
b^4)*d*cos(2*d*sqrt(x) + 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*sin(2*d*sqrt(x
))^2 + (a^12 + 2*a^10*b^2 + a^8*b^4)*d*sin(2*d*sqrt(x) + 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*cos(
2*d*sqrt(x)) + 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*sin(2*d*sqrt(x)) + (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 - 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*cos(2*d*sqrt(x)) - (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*sin(2*d*sqrt(x)) - (a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b
^6 + a^4*b^8)*d)*cos(2*d*sqrt(x) + 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*cos(2*d*sqrt(x)) + ((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*sin(2*d*sqrt(x)))*sin(2*d*sqrt(x) + 2*c))*x*integrate(-2*(2*(a^5
*b*d*sin(2*d*sqrt(x) + 2*c) - (a*b^5*sin(2*c) + 2*(a^4*b^2 + a^2*b^4)*cos(2*c))*d*cos(2*d*sqrt(x)) - (a*b^5*co
s(2*c) - 2*(a^4*b^2 + a^2*b^4)*sin(2*c))*d*sin(2*d*sqrt(x)))*x - (a^4*b^2*sin(2*d*sqrt(x) + 2*c) - (b^6*sin(2*
c) + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*sqrt(x)) - (b^6*cos(2*c) - 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*sq
rt(x)))*sqrt(x))/((a^8*d*cos(2*d*sqrt(x) + 2*c)^2 + a^8*d*sin(2*d*sqrt(x) + 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*cos(2*d*sqrt(x))^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*
sin(2*d*sqrt(x))^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*cos(2*d*sqrt(x)) + 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*sin(2*d*sqrt(x)) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d - 2*((a^4*b^4*cos(2*c) - 2*(a
^7*b + a^5*b^3)*sin(2*c))*d*cos(2*d*sqrt(x)) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*sin(2*d*sqr
t(x)) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d)*cos(2*d*sqrt(x) + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos
(2*c))*d*cos(2*d*sqrt(x)) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*sin(2*d*sqrt(x)))*sin(2*d*sqrt
(x) + 2*c))*x^2), 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*cos(2*d*sqrt(x))^2 + (a^10 - a^8*b^2)*d*cos(2*d*sqrt(x) +
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)*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*sin(2*d*sqrt(x))^2 + (a^10 - a^8*b^2)*d*sin(2*d*sqrt(x) + 2*c)^2 - 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*cos(2*d*sqr
t(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*sin(2*d*sqrt(x)) + (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 - 2*(((a^6*b^4 - a^4*b^6
)*cos(2*c) - 2*(a^9*b - a^5*b^5)*sin(2*c))*d*cos(2*d*sqrt(x)) - (2*(a^9*b - a^5*b^5)*cos(2*c) + (a^6*b^4 - a^4
*b^6)*sin(2*c))*d*sin(2*d*sqrt(x)) - (a^10 + a^8*b^2 - a^6*b^4 - a^4*b^6)*d)*cos(2*d*sqrt(x) + 2*c) - 2*((2*(a
^9*b - a^5*b^5)*cos(2*c) + (a^6*b^4 - a^4*b^6)*sin(2*c))*d*cos(2*d*sqrt(x)) + ((a^6*b^4 - a^4*b^6)*cos(2*c) -
2*(a^9*b - a^5*b^5)*sin(2*c))*d*sin(2*d*sqrt(x)))*sin(2*d*sqrt(x) + 2*c))*x*log(x) - 4*((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*sqrt(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*sqrt(x)) - (a^8*b^2 + 2*a
^6*b^4 + a^4*b^6)*sin(2*d*sqrt(x) + 2*c))*sqrt(x))/((((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*cos(2*d*sqrt(x))^2 + (a^12 + 2*a^10*b^2 + a^8*b^4)*d*cos(2*d*sqrt(x) + 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*sin(2*d*sqrt(x))^2 + (a^12 + 2*a^10*b^2 + a^8*b^4)*d*sin(2*d*sqrt(x) + 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*cos(2*d*sqrt(x)) + 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*sin(
2*d*sqrt(x)) + (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 - 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*cos(2*d*sqrt(x)) -
(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*sin(2*d*
sqrt(x)) - (a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8)*d)*cos(2*d*sqrt(x) + 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*cos(2*d*sqrt(x)) + ((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*sin(2*d*sqrt(
x)))*sin(2*d*sqrt(x) + 2*c))*x)

Giac [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]

[In]

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

[Out]

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