\(\int \frac {a+b \tan (e+f x)}{c+d x} \, dx\) [42]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\text {Int}\left (\frac {a+b \tan (e+f x)}{c+d x},x\right ) \]

[Out]

Unintegrable((a+b*tan(f*x+e))/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int \frac {a+b \tan (e+f x)}{c+d x} \, dx \]

[In]

Int[(a + b*Tan[e + f*x])/(c + d*x),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int \frac {a+b \tan (e+f x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.78 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{d x + c} \,d x } \]

[In]

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

[Out]

(2*b*d*integrate(sin(2*f*x + 2*e)/((d*x + c)*cos(2*f*x + 2*e)^2 + (d*x + c)*sin(2*f*x + 2*e)^2 + d*x + 2*(d*x
+ c)*cos(2*f*x + 2*e) + c), x) + a*log(d*x + c))/d

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \tan (e+f x)}{c+d x} \, dx=\int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{c+d\,x} \,d x \]

[In]

int((a + b*tan(e + f*x))/(c + d*x),x)

[Out]

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