3.457 \(\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx\)

Optimal. Leaf size=37 \[ \text{Unintegrable}\left (\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^m}{\sqrt{a+b \cos (e+f x)}},x\right ) \]

[Out]

Unintegrable[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]

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Rubi [A]  time = 0.123047, 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 = {} \[ \int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]],x]

[Out]

Defer[Int][((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]

Rubi steps

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

Mathematica [A]  time = 7.96176, size = 0, normalized size = 0. \[ \int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]],x]

[Out]

Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]

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Maple [A]  time = 0.464, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c\cos \left ( fx+e \right ) \right ) ^{m} \left ( A+B\cos \left ( fx+e \right ) \right ){\frac{1}{\sqrt{a+b\cos \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(f*x+e))^m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))^(1/2),x)

[Out]

int((c*cos(f*x+e))^m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*(c*cos(f*x + e))^m/sqrt(b*cos(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 (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{\sqrt{b \cos \left (f x + e\right ) + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((B*cos(f*x + e) + A)*(c*cos(f*x + e))^m/sqrt(b*cos(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))**m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))**(1/2),x)

[Out]

Integral((c*cos(e + f*x))**m*(A + B*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(A+B*cos(f*x+e))/(a+b*cos(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((B*cos(f*x + e) + A)*(c*cos(f*x + e))^m/sqrt(b*cos(f*x + e) + a), x)