\(\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\) [214]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\tan (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[Tan[a + b*x]/(c + d*x)^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.48 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Tan[a + b*x]/(c + d*x)^2,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin {\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {\tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (a+b\,x\right )}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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