\(\int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx\) [252]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 19.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\tan {\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 494, normalized size of antiderivative = 24.70 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

2*(cos(2*b*x + 2*a)*cos(b*x + a) + 2*(b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*
cos(2*b*x + 2*a)^2 + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*sin(2*b*x + 2*a)^2 + 2*(b*d^3*x^2 + 2*b*c*d^2*x + b*c
^2*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^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*cos(2*
b*x + 2*a)^2 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*sin(2*b*x + 2*a)^2 + 2*(b*d^3*x^3 + 3*b*c*d^2
*x^2 + 3*b*c^2*d*x + b*c^3)*cos(2*b*x + 2*a)), x) + 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))

Giac [N/A]

Not integrable

Time = 9.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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