\(\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx\) [274]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\text {Int}\left ((a+b \sec (c+d x))^n \sin ^2(c+d x),x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^n*sin(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 21, 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 = {} \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx \]

[In]

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

[Out]

Defer[Int][(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2, x]

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

Mathematica [N/A]

Not integrable

Time = 6.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

integral(-(cos(d*x + c)^2 - 1)*(b*sec(d*x + c) + a)^n, x)

Sympy [N/A]

Not integrable

Time = 37.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \sin ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx=\int {\sin \left (c+d\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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