\(\int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx\) [297]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=-\text {Int}\left ((c+d x)^m \sec (a+b x),x\right )+\text {Int}\left ((c+d x)^m \sec ^3(a+b x),x\right ) \]

[Out]

-Unintegrable((d*x+c)^m*sec(b*x+a),x)+Unintegrable((d*x+c)^m*sec(b*x+a)^3,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 22, 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 = {} \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx \]

[In]

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

[Out]

-Defer[Int][(c + d*x)^m*Sec[a + b*x], x] + Defer[Int][(c + d*x)^m*Sec[a + b*x]^3, x]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^m \sec (a+b x) \, dx+\int (c+d x)^m \sec ^3(a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 51.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int \left (c + d x\right )^{m} \tan ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 28.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx=\int \frac {{\mathrm {tan}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m}{\cos \left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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