3.1522 \(\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx\)

Optimal. Leaf size=38 \[ \text {Int}\left (\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} (a+b \sin (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 36.29, size = 0, normalized size = 0.00 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) + c \cos \left (f x + e\right )^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {1}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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