\(\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx\) [251]

Optimal result

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

[Out]

CannotIntegrate(sec(b*x+a)*tan(b*x+a)/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\tan {\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 349, normalized size of antiderivative = 17.45 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

2*(cos(2*b*x + 2*a)*cos(b*x + a) + (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) + sin(2*b*x + 2*a)*sin(b*x + a) + 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 = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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