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

Optimal. Leaf size=60 \[ (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Unintegrable}\left (\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^{-m}}{\sqrt{a+b \cos (e+f x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.241677, 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 = {} \[ \int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx &=\left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \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 = 8.2039, size = 0, normalized size = 0. \[ \int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.715, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}{\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((A+B*cos(f*x+e))*(c*sec(f*x+e))^m/(a+b*cos(f*x+e))^(1/2),x)

[Out]

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

[Out]

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

[Out]

integral((B*cos(f*x + e) + A)*(c*sec(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 \sec{\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((A+B*cos(f*x+e))*(c*sec(f*x+e))**m/(a+b*cos(f*x+e))**(1/2),x)

[Out]

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

[Out]

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