\(\int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx\) [277]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=4 \text {Int}\left (\frac {\csc ^2(2 a+2 b x)}{(c+d x)^2},x\right ) \]

[Out]

4*Unintegrable(csc(2*b*x+2*a)^2/(d*x+c)^2,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 = {} \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {\csc ^2(2 a+2 b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.80 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 1024, normalized size of antiderivative = 42.67 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

2*(2*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*a)^2 + (b*d^3*x^2
+ 2*b*c*d^2*x + b*c^2*d)*sin(4*b*x + 4*a)^2 - 2*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*a))*integrat
e(sin(2*b*x + 2*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) + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d + (b*d^
3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*a)^2 + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*sin(4*b*x + 4*a)^2 - 2
*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*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) - (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*a)^2 + (b*d^3*x^2
+ 2*b*c*d^2*x + b*c^2*d)*sin(4*b*x + 4*a)^2 - 2*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cos(4*b*x + 4*a))*integrat
e(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*sin(4*b*x + 4*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(4*b*x + 4*a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(4*b*x + 4*a)^2 - 2*(b*d^2*
x^2 + 2*b*c*d*x + b*c^2)*cos(4*b*x + 4*a))

Giac [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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