\(\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx\) [1521]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {Int}\left (\cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n,x\right ) \]

[Out]

Unintegrable(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 33, 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 = {} \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \]

[In]

Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]

[Out]

Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]

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

Mathematica [N/A]

Not integrable

Time = 5.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \]

[In]

Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]

[Out]

Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]

Maple [N/A] (verified)

Not integrable

Time = 0.74 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((b*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e)^2, x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+b*sin(f*x+e))**m*(c+d*sin(f*x+e))**n,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 7.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e)^2, x)

Giac [N/A]

Not integrable

Time = 137.94 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e)^2, x)

Mupad [N/A]

Not integrable

Time = 54.63 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

[In]

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

[Out]

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