\(\int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx\) [238]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 677, normalized size of antiderivative = 30.77 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 5.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 24.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {1}{\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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