\(\int \frac {1}{x^2 (a+b \tan (c+d \sqrt [3]{x}))^2} \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx \]

[In]

Int[1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2),x]

[Out]

Defer[Int][1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 119.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx \]

[In]

Integrate[1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2),x]

[Out]

Integrate[1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.67 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {1}{x^{2} {\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )}^{2}}d x\]

[In]

int(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

[Out]

int(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^2*tan(d*x^(1/3) + c)^2 + 2*a*b*x^2*tan(d*x^(1/3) + c) + a^2*x^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/x**2/(a+b*tan(c+d*x**(1/3)))**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 15.40 (sec) , antiderivative size = 2524, normalized size of antiderivative = 126.20 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="maxima")

[Out]

((a^8*d*cos(2*d*x^(1/3) + 2*c)^2 + a^8*d*sin(2*d*x^(1/3) + 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*x^(1/3))^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*x^(1/3))^
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*x^
(1/3)) + 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*x^(1/3)) + (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)*s
in(2*c))*d*cos(2*d*x^(1/3)) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*sin(2*d*x^(1/3)) - (a^8 + 2*
a^6*b^2 + a^4*b^4)*d)*cos(2*d*x^(1/3) + 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*
x^(1/3)) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*sin(2*d*x^(1/3)))*sin(2*d*x^(1/3) + 2*c))*x^2*i
ntegrate(-4*((a^5*b*d*sin(2*d*x^(1/3) + 2*c) - (a*b^5*sin(2*c) + 2*(a^4*b^2 + a^2*b^4)*cos(2*c))*d*cos(2*d*x^(
1/3)) - (a*b^5*cos(2*c) - 2*(a^4*b^2 + a^2*b^4)*sin(2*c))*d*sin(2*d*x^(1/3)))*x - 2*(a^4*b^2*sin(2*d*x^(1/3) +
2*c) - (b^6*sin(2*c) + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^(1/3)) - (b^6*cos(2*c) - 2*(a^3*b^3 + a*b^5)*s
in(2*c))*sin(2*d*x^(1/3)))*x^(2/3))/((a^8*d*cos(2*d*x^(1/3) + 2*c)^2 + a^8*d*sin(2*d*x^(1/3) + 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*x^(1/3))^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*x^(1/3))^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*x^(1/3)) + 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*x^(1/3)) + (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*x^(1/3)) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(
2*c))*d*sin(2*d*x^(1/3)) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d)*cos(2*d*x^(1/3) + 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*x^(1/3)) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*sin(2*d*x^(
1/3)))*sin(2*d*x^(1/3) + 2*c))*x^3), x) - ((a^6 + a^4*b^2)*d*cos(2*d*x^(1/3) + 2*c)^2 + (a^6 + a^4*b^2)*d*sin(
2*d*x^(1/3) + 2*c)^2 - ((4*a^4*b^2 + 5*a^2*b^4 - b^6)*cos(2*c) - 2*(a^5*b - 2*a*b^5)*sin(2*c))*d*cos(2*d*x^(1/
3)) + (2*(a^5*b - 2*a*b^5)*cos(2*c) + (4*a^4*b^2 + 5*a^2*b^4 - b^6)*sin(2*c))*d*sin(2*d*x^(1/3)) + (a^6 + a^4*
b^2 - a^2*b^4 - b^6)*d - (((a^2*b^4 + b^6)*cos(2*c) - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*sin(2*c))*d*cos(2*d*x^(1/3
)) - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*c) + (a^2*b^4 + b^6)*sin(2*c))*d*sin(2*d*x^(1/3)) - (2*a^6 + 2*a^4*b
^2 + 3*a^2*b^4 + b^6)*d)*cos(2*d*x^(1/3) + 2*c) + (2*a^5*b*d - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*c) + (a^2*
b^4 + b^6)*sin(2*c))*d*cos(2*d*x^(1/3)) - ((a^2*b^4 + b^6)*cos(2*c) - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*sin(2*c))*
d*sin(2*d*x^(1/3)))*sin(2*d*x^(1/3) + 2*c))*x + 6*(a^4*b^2*sin(2*d*x^(1/3) + 2*c) - (b^6*sin(2*c) + 2*(a^3*b^3
+ a*b^5)*cos(2*c))*cos(2*d*x^(1/3)) - (b^6*cos(2*c) - 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^(1/3)))*x^(2/3)
)/((a^8*d*cos(2*d*x^(1/3) + 2*c)^2 + a^8*d*sin(2*d*x^(1/3) + 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*x^(1/3))^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*x^(1/3)
)^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*
x^(1/3)) + 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*x^(1/3)) + (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*x^(1/3)) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*sin(2*d*x^(1/3)) - (a^8 +
2*a^6*b^2 + a^4*b^4)*d)*cos(2*d*x^(1/3) + 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*x^(1/3)) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*sin(2*d*x^(1/3)))*sin(2*d*x^(1/3) + 2*c))*x^2
)

Giac [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2} \,d x \]

[In]

int(1/(x^2*(a + b*tan(c + d*x^(1/3)))^2),x)

[Out]

int(1/(x^2*(a + b*tan(c + d*x^(1/3)))^2), x)