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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\cos (c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0457507, 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 = {} \[ \int \frac{(e+f x)^m \cos (c+d x)}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.29973, size = 0, normalized size = 0. \[ \int \frac{(e+f x)^m \cos (c+d x)}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m}\cos \left ( dx+c \right ) }{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x)**m*cos(c + d*x)/(a + b*sin(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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