3.1526 \(\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx\)

Optimal. Leaf size=35 \[ \text{Unintegrable}\left (\frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.140571, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.03357, size = 0, normalized size = 0. \[ \int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.697, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{a+b\sin \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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