\(\int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx\) [270]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.17 (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 = {} \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 19.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 603, normalized size of antiderivative = 27.41 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

(2*cos(2*b*x + 2*a)*cos(b*x + a) + 2*(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((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^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*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(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x +
2*(d*x + c)*cos(b*x + a) + c), x) + (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(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin
(b*x + a)^2 + d*x - 2*(d*x + c)*cos(b*x + a) + c), x) + 2*sin(2*b*x + 2*a)*sin(b*x + a) + 2*cos(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 = 4.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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