\(\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx\) [1522]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\text {Int}\left (\cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3},x\right ) \]

[Out]

Unintegrable(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x)

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 35, 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 = {} \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \]

[In]

Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3),x]

[Out]

Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), 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))^{4/3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 12.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \]

[In]

Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3),x]

[Out]

Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}d x\]

[In]

int(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x)

[Out]

int(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x)

Fricas [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x, algorithm="fricas")

[Out]

integral((d*cos(f*x + e)^2*sin(f*x + e) + c*cos(f*x + e)^2)*(d*sin(f*x + e) + c)^(1/3)*(b*sin(f*x + e) + a)^m,
x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+b*sin(f*x+e))**m*(c+d*sin(f*x+e))**(4/3),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 3.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^(4/3)*(b*sin(f*x + e) + a)^m*cos(f*x + e)^2, x)

Giac [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(4/3),x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^(4/3)*(b*sin(f*x + e) + a)^m*cos(f*x + e)^2, x)

Mupad [N/A]

Not integrable

Time = 27.52 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]

[In]

int(cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(4/3),x)

[Out]

int(cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(4/3), x)