\(\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx\) [282]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.27 (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 = {} \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 23.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.32 (sec) , antiderivative size = 3697, normalized size of antiderivative = 154.04 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 119.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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