\(\int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx\) [321]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.26 (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 = {} \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 18.72 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 14.81 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 7.40 (sec) , antiderivative size = 4747, normalized size of antiderivative = 197.79 \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 25.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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