\(\int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx\) [226]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\text {Int}\left (\frac {1}{1+\tan ^{2 \sqrt {505}}(x)},x\right ) \]

[Out]

Unintegrable(1/(tan(x)^(2*505^(1/2))+1),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx \]

[In]

Int[(1 + Tan[x]^(2*Sqrt[505]))^(-1),x]

[Out]

Defer[Int][(1 + Tan[x]^(2*Sqrt[505]))^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.78 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx \]

[In]

Integrate[(1 + Tan[x]^(2*Sqrt[505]))^(-1),x]

[Out]

Integrate[(1 + Tan[x]^(2*Sqrt[505]))^(-1), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

\[\int \frac {1}{\tan \left (x \right )^{2 \sqrt {505}}+1}d x\]

[In]

int(1/(tan(x)^(2*505^(1/2))+1),x)

[Out]

int(1/(tan(x)^(2*505^(1/2))+1),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int { \frac {1}{\tan \left (x\right )^{2 \, \sqrt {505}} + 1} \,d x } \]

[In]

integrate(1/(tan(x)^(2*505^(1/2))+1),x, algorithm="fricas")

[Out]

integral(1/(tan(x)^(2*sqrt(505)) + 1), x)

Sympy [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int \frac {1}{\tan ^{2 \sqrt {505}}{\left (x \right )} + 1}\, dx \]

[In]

integrate(1/(tan(x)**(2*505**(1/2))+1),x)

[Out]

Integral(1/(tan(x)**(2*sqrt(505)) + 1), x)

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 1038, normalized size of antiderivative = 74.14 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int { \frac {1}{\tan \left (x\right )^{2 \, \sqrt {505}} + 1} \,d x } \]

[In]

integrate(1/(tan(x)^(2*505^(1/2))+1),x, algorithm="maxima")

[Out]

-(-1)^(sqrt(101)*sqrt(5))*integrate(((-1)^(sqrt(101)*sqrt(5))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) +
1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1))^2*e^(2*sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*c
os(x) + 1) + 2*sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)) + (-1)^(sqrt(101)*sqrt(5))*e^(2*sqrt
(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 2*sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 - 2*cos(x)
+ 1))*sin(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) + 1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1))
^2 + cos(2*sqrt(101)*sqrt(5)*arctan2(sin(2*x), cos(2*x) + 1))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) +
1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1))*e^(sqrt(101)*sqrt(5)*log(cos(2*x)^2 + sin(2*x)^2 + 2*c
os(2*x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + si
n(x)^2 - 2*cos(x) + 1)) + e^(sqrt(101)*sqrt(5)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) + sqrt(101)*sqrt(
5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1))*sin(2*
sqrt(101)*sqrt(5)*arctan2(sin(2*x), cos(2*x) + 1))*sin(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) + 1) - 2*sqr
t(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1)))/(2*(-1)^(sqrt(101)*sqrt(5))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(
2*x), cos(2*x) + 1))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) + 1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x),
-cos(x) + 1))*e^(sqrt(101)*sqrt(5)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) + sqrt(101)*sqrt(5)*log(cos(
x)^2 + sin(x)^2 + 2*cos(x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)) + (-1)^(2*sqrt(10
1)*sqrt(5))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) + 1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x)
+ 1))^2*e^(2*sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 2*sqrt(101)*sqrt(5)*log(cos(x)^2 + si
n(x)^2 - 2*cos(x) + 1)) + 2*(-1)^(sqrt(101)*sqrt(5))*e^(sqrt(101)*sqrt(5)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(
2*x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x
)^2 - 2*cos(x) + 1))*sin(2*sqrt(101)*sqrt(5)*arctan2(sin(2*x), cos(2*x) + 1))*sin(2*sqrt(101)*sqrt(5)*arctan2(
sin(x), cos(x) + 1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1)) + (-1)^(2*sqrt(101)*sqrt(5))*e^(2*sqrt
(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 2*sqrt(101)*sqrt(5)*log(cos(x)^2 + sin(x)^2 - 2*cos(x)
+ 1))*sin(2*sqrt(101)*sqrt(5)*arctan2(sin(x), cos(x) + 1) - 2*sqrt(101)*sqrt(5)*arctan2(sin(x), -cos(x) + 1))
^2 + (cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(2*sqrt(101)*sqrt(5))*cos(2*sqrt(101)*sqrt(5)*arctan2(sin(2*x)
, cos(2*x) + 1))^2 + (cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(2*sqrt(101)*sqrt(5))*sin(2*sqrt(101)*sqrt(5)*
arctan2(sin(2*x), cos(2*x) + 1))^2), x) + x

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int { \frac {1}{\tan \left (x\right )^{2 \, \sqrt {505}} + 1} \,d x } \]

[In]

integrate(1/(tan(x)^(2*505^(1/2))+1),x, algorithm="giac")

[Out]

integrate(1/(tan(x)^(2*sqrt(505)) + 1), x)

Mupad [N/A]

Not integrable

Time = 15.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\tan ^{2 \sqrt {505}}(x)} \, dx=\int \frac {1}{{\mathrm {tan}\left (x\right )}^{2\,\sqrt {505}}+1} \,d x \]

[In]

int(1/(tan(x)^(2*505^(1/2)) + 1),x)

[Out]

int(1/(tan(x)^(2*505^(1/2)) + 1), x)