\(\int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx\) [326]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=8 \text {Int}\left (\frac {\csc ^3(2 a+2 b x)}{c+d x},x\right ) \]

[Out]

8*Unintegrable(csc(2*b*x+2*a)^3/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = 8 \int \frac {\csc ^3(2 a+2 b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 21.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 20.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.93 (sec) , antiderivative size = 2410, normalized size of antiderivative = 100.42 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

(2*(2*(b*d*x + b*c)*cos(6*b*x + 6*a) + 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - d*sin(6*b*x + 6*a) + d*sin(2*b*x + 2
*a))*cos(8*b*x + 8*a) + 4*(b*d*x + b*c - 2*(b*d*x + b*c)*cos(4*b*x + 4*a) - d*sin(4*b*x + 4*a))*cos(6*b*x + 6*
a) - 4*(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) + 4*(b*d*x + b*c)*cos(2*b*x +
2*a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(8*b*x + 8*a)^2 + 4*(b^
2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)
^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x
+ b^2*c^2)*sin(4*b*x + 4*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^
2)*cos(4*b*x + 4*a))*cos(8*b*x + 8*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a))*integrate(2*
(2*b^2*d^2*x^2 + 4*b^2*c*d*x + 2*b^2*c^2 + d^2)*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(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(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)*cos(2*b*x + 2*a)), x) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^
2)*cos(8*b*x + 8*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*
x + b^2*c^2)*sin(8*b*x + 8*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) +
4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d
^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a))*cos(8*b*x + 8*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*c
os(4*b*x + 4*a))*integrate((2*b^2*d^2*x^2 + 4*b^2*c*d*x + 2*b^2*c^2 + d^2)*sin(b*x + 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(b*x + 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(b*x + 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)*cos(b*x + a)), x) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d
*x + b^2*c^2)*cos(8*b*x + 8*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + (b^2*d^2*x^2 +
2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)*sin(4*b*
x + 4*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2
- 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a))*cos(8*b*x + 8*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2)*cos(4*b*x + 4*a))*integrate((2*b^2*d^2*x^2 + 4*b^2*c*d*x + 2*b^2*c^2 + d^2)*sin(b*x + 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(
b*x + 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(b*x + 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)*cos(b*x + a)), x) + 2*(d*cos(6*b*x + 6*a) - d*cos(2*b*x + 2*a) + 2*(b*d*
x + b*c)*sin(6*b*x + 6*a) + 2*(b*d*x + b*c)*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + 2*(2*d*cos(4*b*x + 4*a) - 4*(
b*d*x + b*c)*sin(4*b*x + 4*a) - d)*sin(6*b*x + 6*a) + 4*(d*cos(2*b*x + 2*a) - 2*(b*d*x + b*c)*sin(2*b*x + 2*a)
)*sin(4*b*x + 4*a) + 2*d*sin(2*b*x + 2*a))/(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2)*cos(8*b*x + 8*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + (b^2*d^2*x^2 + 2*b
^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(8*b*x + 8*a)*sin(4*b*x +
4*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2
*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a))*cos(8*b*x + 8*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2
*c^2)*cos(4*b*x + 4*a))

Giac [N/A]

Not integrable

Time = 3.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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