\(\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx\) [283]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.25 (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 = {} \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 29.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 14.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 11.08 (sec) , antiderivative size = 4757, normalized size of antiderivative = 198.21 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(csc(b*x+a)^3*sec(b*x+a)^2/(d*x+c)^2,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) - 2*d*sin(5*b*x + 5*a) + 2*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) - 2*d*sin(4*b*x + 4*a) - 2*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) - 2*d*s
in(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) + 2*d*sin(b*x + a))*cos(2*b*x + 2*a) + 3*(b*d*x + b*c)*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(3/2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 2*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(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))*integr
ate(3/2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 2*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) + 4*(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))*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^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(2*b*x + 2*a)^
2 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*sin(2*b*x + 2*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(2*b*x + 2*a)), x) + (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))*sin(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)*sin(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))*sin(4*b*x +
4*a) + (2*d*cos(b*x + a) - 3*(b*d*x + b*c)*sin(b*x + a))*sin(2*b*x + 2*a) + 2*d*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. \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 26.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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