\(\int \frac {\tan (a+b x)}{c+d x} \, dx\) [213]

Optimal result

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

[Out]

Unintegrable(tan(b*x+a)/(d*x+c),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 = {} \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int \frac {\tan (a+b x)}{c+d x} \, dx \]

[In]

Int[Tan[a + b*x]/(c + d*x),x]

[Out]

Defer[Int][Tan[a + b*x]/(c + d*x), x]

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

Mathematica [N/A]

Not integrable

Time = 5.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int \frac {\tan (a+b x)}{c+d x} \, dx \]

[In]

Integrate[Tan[a + b*x]/(c + d*x),x]

[Out]

Integrate[Tan[a + b*x]/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {\tan (a+b x)}{c+d x} \, dx=\int \frac {\sin \left (a+b\,x\right )}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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