\(\int (c+d x)^m \tan (a+b x) \, dx\) [208]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int (c+d x)^m \tan (a+b x) \, dx=\int (c+d x)^m \tan (a+b x) \, dx \]

[In]

Int[(c + d*x)^m*Tan[a + b*x],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.72 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int (c+d x)^m \tan (a+b x) \, dx=\int (c+d x)^m \tan (a+b x) \, dx \]

[In]

Integrate[(c + d*x)^m*Tan[a + b*x],x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int (c+d x)^m \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 30.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (c+d x)^m \tan (a+b x) \, dx=\int \left (c + d x\right )^{m} \sin {\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int (c+d x)^m \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 24.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int (c+d x)^m \tan (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m}{\cos \left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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