\(\int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx\) [245]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 18.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 9.09 (sec) , antiderivative size = 2702, normalized size of antiderivative = 122.82 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 - 2*((b*d*x + b*c)*cos(2*b*x + 2*a)
- d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3
*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2
*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*
b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*
b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 +
2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x
+ b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*c
os(2*b*x + 2*a))*integrate((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 3*d^2)*sin(b*x + a)/(b^2*d^4*x^4 + 4*b^2*c*d
^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*
b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)^2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b
^2*c^4)*sin(b*x + a)^2 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b
*x + a)), x) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b
^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b
*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 +
3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^
2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 -
2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x
^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*integrate((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c
^2 + 3*d^2)*sin(b*x + a)/(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d
^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)^2 + (b^2*d^4*x^4 + 4*b^2*
c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*sin(b*x + a)^2 - 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6
*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)), x) - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d
*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 +
3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*
x + b^2*c^3)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)
*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^
3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3
)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x +
2*a))*integrate(sin(2*b*x + 2*a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(2*b*x + 2*a)^2 + (d^2*x^2
+ 2*c*d*x + c^2)*sin(2*b*x + 2*a)^2 + c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(2*b*x + 2*a)), x) - 2*(d*cos(2*b*
x + 2*a) + (b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(4*b*x + 4*a) - 2*d*sin(2*b*x + 2*a))/(b^2*d^3*x^3 + 3*b^2*c
*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a
)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*
d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2
*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*
x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3
*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x
+ b^2*c^3)*cos(2*b*x + 2*a))

Giac [F(-1)]

Timed out. \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 24.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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