\(\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx\) [289]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.65 (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 = {} \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 23.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 11.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 1884, normalized size of antiderivative = 94.20 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(csc(b*x+a)^3*sec(b*x+a)^2/x^2,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) - 2*sin(5*b*x + 5*a) + 2*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 + 2*sin(4*b*x + 4*a) + 2*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) - 2*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) + 2*sin(b*x + a))*cos(2*b*x + 2*a) + (b^2*x^3*cos(6*b*x + 6*a)^2 + b^2*x^3*cos(4*b*x +
4*a)^2 + b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(6*b*x + 6*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 2*b^2*x^3*sin(
4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^3*sin(2*b*x + 2*a)^2 - 2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 - 2*(b^2*x^3
*cos(4*b*x + 4*a) + b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(6*b*x + 6*a) + 2*(b^2*x^3*cos(2*b*x + 2*a) - b^2*x
^3)*cos(4*b*x + 4*a) - 2*(b^2*x^3*sin(4*b*x + 4*a) + b^2*x^3*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*integrate(3/2
*(b^2*x^2 + 2)*sin(b*x + a)/(b^2*x^4*cos(b*x + a)^2 + b^2*x^4*sin(b*x + a)^2 + 2*b^2*x^4*cos(b*x + a) + b^2*x^
4), x) + (b^2*x^3*cos(6*b*x + 6*a)^2 + b^2*x^3*cos(4*b*x + 4*a)^2 + b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(6
*b*x + 6*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 2*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^3*sin(2*b*x +
2*a)^2 - 2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 - 2*(b^2*x^3*cos(4*b*x + 4*a) + b^2*x^3*cos(2*b*x + 2*a) - b^2*
x^3)*cos(6*b*x + 6*a) + 2*(b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(4*b*x + 4*a) - 2*(b^2*x^3*sin(4*b*x + 4*a)
+ b^2*x^3*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*integrate(3/2*(b^2*x^2 + 2)*sin(b*x + a)/(b^2*x^4*cos(b*x + a)^2
+ b^2*x^4*sin(b*x + a)^2 - 2*b^2*x^4*cos(b*x + a) + b^2*x^4), x) + 4*(b^2*x^3*cos(6*b*x + 6*a)^2 + b^2*x^3*co
s(4*b*x + 4*a)^2 + b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(6*b*x + 6*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 2*b^
2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*x^3*sin(2*b*x + 2*a)^2 - 2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 -
2*(b^2*x^3*cos(4*b*x + 4*a) + b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(6*b*x + 6*a) + 2*(b^2*x^3*cos(2*b*x + 2*
a) - b^2*x^3)*cos(4*b*x + 4*a) - 2*(b^2*x^3*sin(4*b*x + 4*a) + b^2*x^3*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*int
egrate((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^3*cos(2*b*x + 2*a)^
2 + b*x^3*sin(2*b*x + 2*a)^2 + 2*b*x^3*cos(2*b*x + 2*a) + b*x^3), 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) + 2*cos(5*b*x + 5*a) - 2*cos(b*x + a))*sin(6*b*x + 6*a) - (3*b*x*sin(4*b*x + 4*
a) + 3*b*x*sin(2*b*x + 2*a) - 2*cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 2)*sin(5*b*x + 5*a) + (2*b*x*sin(3*b*x
+ 3*a) - 3*b*x*sin(b*x + a) + 2*cos(b*x + a))*sin(4*b*x + 4*a) - (3*b*x*sin(b*x + a) - 2*cos(b*x + a))*sin(2*
b*x + 2*a) + 2*sin(b*x + a))/(b^2*x^3*cos(6*b*x + 6*a)^2 + b^2*x^3*cos(4*b*x + 4*a)^2 + b^2*x^3*cos(2*b*x + 2*
a)^2 + b^2*x^3*sin(6*b*x + 6*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 2*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) +
b^2*x^3*sin(2*b*x + 2*a)^2 - 2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 - 2*(b^2*x^3*cos(4*b*x + 4*a) + b^2*x^3*cos
(2*b*x + 2*a) - b^2*x^3)*cos(6*b*x + 6*a) + 2*(b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(4*b*x + 4*a) - 2*(b^2*x
^3*sin(4*b*x + 4*a) + b^2*x^3*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))

Giac [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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