3.5.57 \(\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx\) [457]

Optimal. Leaf size=38 \[ \text {Int}\left (\frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}},x\right ) \]

[Out]

Unintegrable((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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Rubi [A]
time = 0.07, 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 \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 9.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx \end {gather*}

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 = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (c \cos \left (f x +e \right )\right )^{m} \left (A +B \cos \left (f x +e \right )\right )}{\sqrt {a +b \cos \left (f x +e \right )}}\, dx\]

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 = 0.00, 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((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 = 0.00, 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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