\(\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx\) [288]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\text {Int}\left (\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x},x\right ) \]

[Out]

CannotIntegrate(csc(b*x+a)^3*sec(b*x+a)^2/x,x)

Rubi [N/A]

Not integrable

Time = 0.59 (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 = {} \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx \]

[In]

Int[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x,x]

[Out]

Defer[Int][(Csc[a + b*x]^3*Sec[a + b*x]^2)/x, x]

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

Mathematica [N/A]

Not integrable

Time = 54.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx \]

[In]

Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x,x]

[Out]

Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integral(csc(b*x + a)^3*sec(b*x + a)^2/x, x)

Sympy [N/A]

Not integrable

Time = 8.70 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(csc(a + b*x)**3*sec(a + b*x)**2/x, x)

Maxima [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 1872, normalized size of antiderivative = 93.60 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

(2*b*x*sin(3*b*x + 3*a)*sin(2*b*x + 2*a) + 3*b*x*cos(b*x + a) + (3*b*x*cos(5*b*x + 5*a) - 2*b*x*cos(3*b*x + 3*
a) + 3*b*x*cos(b*x + a) - sin(5*b*x + 5*a) + sin(b*x + a))*cos(6*b*x + 6*a) - (3*b*x*cos(4*b*x + 4*a) + 3*b*x*
cos(2*b*x + 2*a) - 3*b*x + sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*cos(5*b*x + 5*a) + (2*b*x*cos(3*b*x + 3*a) - 3
*b*x*cos(b*x + a) - sin(b*x + a))*cos(4*b*x + 4*a) + 2*(b*x*cos(2*b*x + 2*a) - b*x)*cos(3*b*x + 3*a) - (3*b*x*
cos(b*x + a) + sin(b*x + a))*cos(2*b*x + 2*a) + (b^2*x^2*cos(6*b*x + 6*a)^2 + b^2*x^2*cos(4*b*x + 4*a)^2 + b^2
*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(6*b*x + 6*a)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 + 2*b^2*x^2*sin(4*b*x + 4*a)
*sin(2*b*x + 2*a) + b^2*x^2*sin(2*b*x + 2*a)^2 - 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 - 2*(b^2*x^2*cos(4*b*x +
4*a) + b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(6*b*x + 6*a) + 2*(b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(4*b*
x + 4*a) - 2*(b^2*x^2*sin(4*b*x + 4*a) + b^2*x^2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*integrate(1/2*(3*b^2*x^2
+ 2)*sin(b*x + a)/(b^2*x^3*cos(b*x + a)^2 + b^2*x^3*sin(b*x + a)^2 + 2*b^2*x^3*cos(b*x + a) + b^2*x^3), x) + (
b^2*x^2*cos(6*b*x + 6*a)^2 + b^2*x^2*cos(4*b*x + 4*a)^2 + b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(6*b*x + 6*a
)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 + 2*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^2*sin(2*b*x + 2*a)^2 -
2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 - 2*(b^2*x^2*cos(4*b*x + 4*a) + b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(6
*b*x + 6*a) + 2*(b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(4*b*x + 4*a) - 2*(b^2*x^2*sin(4*b*x + 4*a) + b^2*x^2*
sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*integrate(1/2*(3*b^2*x^2 + 2)*sin(b*x + a)/(b^2*x^3*cos(b*x + a)^2 + b^2*x
^3*sin(b*x + a)^2 - 2*b^2*x^3*cos(b*x + a) + b^2*x^3), x) + 2*(b^2*x^2*cos(6*b*x + 6*a)^2 + b^2*x^2*cos(4*b*x
+ 4*a)^2 + b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(6*b*x + 6*a)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 + 2*b^2*x^2*si
n(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^2*sin(2*b*x + 2*a)^2 - 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 - 2*(b^2*x
^2*cos(4*b*x + 4*a) + b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(6*b*x + 6*a) + 2*(b^2*x^2*cos(2*b*x + 2*a) - b^2
*x^2)*cos(4*b*x + 4*a) - 2*(b^2*x^2*sin(4*b*x + 4*a) + b^2*x^2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*integrate((
cos(2*b*x + 2*a)*cos(b*x + a) + sin(2*b*x + 2*a)*sin(b*x + a) + cos(b*x + a))/(b*x^2*cos(2*b*x + 2*a)^2 + b*x^
2*sin(2*b*x + 2*a)^2 + 2*b*x^2*cos(2*b*x + 2*a) + b*x^2), x) + (3*b*x*sin(5*b*x + 5*a) - 2*b*x*sin(3*b*x + 3*a
) + 3*b*x*sin(b*x + a) + cos(5*b*x + 5*a) - cos(b*x + a))*sin(6*b*x + 6*a) - (3*b*x*sin(4*b*x + 4*a) + 3*b*x*s
in(2*b*x + 2*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*sin(5*b*x + 5*a) + (2*b*x*sin(3*b*x + 3*a) - 3*b*x*
sin(b*x + a) + cos(b*x + a))*sin(4*b*x + 4*a) - (3*b*x*sin(b*x + a) - cos(b*x + a))*sin(2*b*x + 2*a) + sin(b*x
+ a))/(b^2*x^2*cos(6*b*x + 6*a)^2 + b^2*x^2*cos(4*b*x + 4*a)^2 + b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(6*b
*x + 6*a)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 + 2*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^2*sin(2*b*x + 2
*a)^2 - 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 - 2*(b^2*x^2*cos(4*b*x + 4*a) + b^2*x^2*cos(2*b*x + 2*a) - b^2*x^
2)*cos(6*b*x + 6*a) + 2*(b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(4*b*x + 4*a) - 2*(b^2*x^2*sin(4*b*x + 4*a) +
b^2*x^2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))

Giac [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(csc(b*x + a)^3*sec(b*x + a)^2/x, x)

Mupad [N/A]

Not integrable

Time = 26.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx=\int \frac {1}{x\,{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

int(1/(x*cos(a + b*x)^2*sin(a + b*x)^3),x)

[Out]

int(1/(x*cos(a + b*x)^2*sin(a + b*x)^3), x)