\(\int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx\) [276]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 9.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 687, normalized size of antiderivative = 28.62 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

(2*(b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(4*b*x + 4*a)^2 + (b*d^2*x + b*c*d)*sin(4*b*x + 4*a)^2 - 2*(b*d^2*x
+ b*c*d)*cos(4*b*x + 4*a))*integrate(sin(2*b*x + 2*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) + (b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(4*b*x + 4*a)^2 + (b*d^2*x + b*c*d)*si
n(4*b*x + 4*a)^2 - 2*(b*d^2*x + b*c*d)*cos(4*b*x + 4*a))*integrate(sin(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(b*x + a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(b*x + a)^2 + 2*(b*d^2*
x^2 + 2*b*c*d*x + b*c^2)*cos(b*x + a)), x) - (b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(4*b*x + 4*a)^2 + (b*d^2*
x + b*c*d)*sin(4*b*x + 4*a)^2 - 2*(b*d^2*x + b*c*d)*cos(4*b*x + 4*a))*integrate(sin(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(b*x + a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(b*x + a)^
2 - 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(b*x + a)), x) - 4*sin(4*b*x + 4*a))/(b*d*x + (b*d*x + b*c)*cos(4*b*x
+ 4*a)^2 + (b*d*x + b*c)*sin(4*b*x + 4*a)^2 + b*c - 2*(b*d*x + b*c)*cos(4*b*x + 4*a))

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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