\(\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx\) [643]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=(c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left (\frac {(c \cos (e+f x))^{-m} (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 39.77 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 1.58 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \frac {\left (A +\cos \left (f x +e \right ) B \right ) \left (c \sec \left (f x +e \right )\right )^{m}}{\sqrt {a +b \cos \left (f x +e \right )}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 3.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {\left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\sqrt {a + b \cos {\left (e + f x \right )}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 2.61 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 4.94 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{\sqrt {a+b\,\cos \left (e+f\,x\right )}} \,d x \]

[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)