\(\int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx\) [232]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=2 \text {Int}\left (\frac {\csc (2 a+2 b x)}{c+d x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.72 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {\csc (a+b x) \sec (a+b x)}{c+d x} \, dx=\int \frac {1}{\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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