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

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

[Out]

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

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Rubi [A]
time = 0.11, 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 = {} \begin {gather*} \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \end {gather*}

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 = 78.13, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx \end {gather*}

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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Maple [A]
time = 0.17, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 8569 deep

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}

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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