\(\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx\) [273]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\text {Int}\left ((a+b \sec (c+d x))^n \sin ^4(c+d x),x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^n*sin(d*x+c)^4,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 = {} \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx \]

[In]

Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4,x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4, x]

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx \]

[In]

Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4,x]

[Out]

Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4, x]

Maple [N/A] (verified)

Not integrable

Time = 2.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )^{4}d x\]

[In]

int((a+b*sec(d*x+c))^n*sin(d*x+c)^4,x)

[Out]

int((a+b*sec(d*x+c))^n*sin(d*x+c)^4,x)

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="fricas")

[Out]

integral((cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*(b*sec(d*x + c) + a)^n, x)

Sympy [F(-1)]

Timed out. \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**n*sin(d*x+c)**4,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 3.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^n*sin(d*x + c)^4, x)

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^n*sin(d*x + c)^4, x)

Mupad [N/A]

Not integrable

Time = 18.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx=\int {\sin \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

int(sin(c + d*x)^4*(a + b/cos(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^4*(a + b/cos(c + d*x))^n, x)