\(\int \frac {\tan ^3(a+b x)}{x^2} \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\text {Int}\left (\frac {\tan ^3(a+b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int \frac {\tan ^3(a+b x)}{x^2} \, dx \]

[In]

Int[Tan[a + b*x]^3/x^2,x]

[Out]

Defer[Int][Tan[a + b*x]^3/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan ^3(a+b x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.58 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int \frac {\tan ^3(a+b x)}{x^2} \, dx \]

[In]

Integrate[Tan[a + b*x]^3/x^2,x]

[Out]

Integrate[Tan[a + b*x]^3/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int { \frac {\tan \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(tan(b*x + a)^3/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int \frac {\tan ^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(tan(a + b*x)**3/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 536, normalized size of antiderivative = 44.67 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int { \frac {\tan \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

(4*b*x*cos(2*b*x + 2*a)^2 + 4*b*x*sin(2*b*x + 2*a)^2 + 2*b*x*cos(2*b*x + 2*a) + 2*(b*x*cos(2*b*x + 2*a) - sin(
2*b*x + 2*a))*cos(4*b*x + 4*a) - (b^2*x^3*cos(4*b*x + 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*
x + 4*a)^2 + 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 + 4*b^2*x^3*cos(2*b*x
+ 2*a) + b^2*x^3 + 2*(2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3)*cos(4*b*x + 4*a))*integrate(2*(b^2*x^2 - 3)*sin(2*
b*x + 2*a)/(b^2*x^4*cos(2*b*x + 2*a)^2 + b^2*x^4*sin(2*b*x + 2*a)^2 + 2*b^2*x^4*cos(2*b*x + 2*a) + b^2*x^4), x
) + 2*(b*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))/(b^2*x^3*cos(4*b*x
+ 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x +
2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 + 2*(2*b^2*x^3*cos(2*b*x + 2*a) + b
^2*x^3)*cos(4*b*x + 4*a))

Giac [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int { \frac {\tan \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(tan(b*x + a)^3/x^2, x)

Mupad [N/A]

Not integrable

Time = 2.83 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^3(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{x^2} \,d x \]

[In]

int(tan(a + b*x)^3/x^2,x)

[Out]

int(tan(a + b*x)^3/x^2, x)