\(\int (c (d \sin (e+f x))^p)^n (3+b \sin (e+f x))^m \, dx\) [831]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=(d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \text {Int}\left ((d \sin (e+f x))^{n p} (3+b \sin (e+f x))^m,x\right ) \]

[Out]

(c*(d*sin(f*x+e))^p)^n*Unintegrable((d*sin(f*x+e))^(n*p)*(a+b*sin(f*x+e))^m,x)/((d*sin(f*x+e))^(n*p))

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 27, 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 = {} \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^m \, dx \]

[In]

Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^m,x]

[Out]

((c*(d*Sin[e + f*x])^p)^n*Defer[Int][(d*Sin[e + f*x])^(n*p)*(a + b*Sin[e + f*x])^m, x])/(d*Sin[e + f*x])^(n*p)

Rubi steps \begin{align*} \text {integral}& = \left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} (a+b \sin (e+f x))^m \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx \]

[In]

Integrate[(c*(d*Sin[e + f*x])^p)^n*(3 + b*Sin[e + f*x])^m,x]

[Out]

Integrate[(c*(d*Sin[e + f*x])^p)^n*(3 + b*Sin[e + f*x])^m, x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

\[\int \left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \sin \left (f x +e \right )\right )^{m}d x\]

[In]

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x)

[Out]

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x)

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int { \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral(((d*sin(f*x + e))^p*c)^n*(b*sin(f*x + e) + a)^m, x)

Sympy [N/A]

Not integrable

Time = 23.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

integrate((c*(d*sin(f*x+e))**p)**n*(a+b*sin(f*x+e))**m,x)

[Out]

Integral((c*(d*sin(e + f*x))**p)**n*(a + b*sin(e + f*x))**m, x)

Maxima [N/A]

Not integrable

Time = 5.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int { \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate(((d*sin(f*x + e))^p*c)^n*(b*sin(f*x + e) + a)^m, x)

Giac [N/A]

Not integrable

Time = 6.74 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int { \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate(((d*sin(f*x + e))^p*c)^n*(b*sin(f*x + e) + a)^m, x)

Mupad [N/A]

Not integrable

Time = 10.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^m \, dx=\int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

[In]

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^m,x)

[Out]

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^m, x)