\(\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\) [641]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\frac {2 b B \cos (e+f x) \sqrt {a+b \cos (e+f x)} (c \sec (e+f x))^m \sin (e+f x)}{f (5-2 m)}+\frac {2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left (\frac {(c \cos (e+f x))^{-m} \left (\frac {1}{2} a c \left (2 b B (1-m)+2 a A \left (\frac {5}{2}-m\right )\right )+\frac {1}{2} c \left (b^2 B (3-2 m)+a (2 A b+a B) (5-2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (A b (5-2 m)+2 a B (3-m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{c (5-2 m)} \]

[Out]

2*b*B*cos(f*x+e)*(c*sec(f*x+e))^m*sin(f*x+e)*(a+b*cos(f*x+e))^(1/2)/f/(5-2*m)+2*(c*cos(f*x+e))^m*(c*sec(f*x+e)
)^m*Unintegrable((1/2*a*c*(2*b*B*(1-m)+2*a*A*(5/2-m))+1/2*c*(b^2*B*(3-2*m)+a*(2*A*b+B*a)*(5-2*m))*cos(f*x+e)+1
/2*b*c*(A*b*(5-2*m)+2*a*B*(3-m))*cos(f*x+e)^2)/((c*cos(f*x+e))^m)/(a+b*cos(f*x+e))^(1/2),x)/c/(5-2*m)

Rubi [N/A]

Not integrable

Time = 1.00 (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 = {} \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

[In]

Int[(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

(2*b*B*Cos[e + f*x]*Sqrt[a + b*Cos[e + f*x]]*(c*Sec[e + f*x])^m*Sin[e + f*x])/(f*(5 - 2*m)) + (2*(c*Cos[e + f*
x])^m*(c*Sec[e + f*x])^m*Defer[Int][((a*c*(2*b*B*(1 - m) + 2*a*A*(5/2 - m)))/2 + (c*(b^2*B*(3 - 2*m) + a*(2*A*
b + a*B)*(5 - 2*m))*Cos[e + f*x])/2 + (b*c*(A*b*(5 - 2*m) + 2*a*B*(3 - m))*Cos[e + f*x]^2)/2)/((c*Cos[e + f*x]
)^m*Sqrt[a + b*Cos[e + f*x]]), x])/(c*(5 - 2*m))

Rubi steps \begin{align*} \text {integral}& = \left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \\ & = \frac {2 b B \cos (e+f x) \sqrt {a+b \cos (e+f x)} (c \sec (e+f x))^m \sin (e+f x)}{f (5-2 m)}+\frac {\left (2 (c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int \frac {(c \cos (e+f x))^{-m} \left (\frac {1}{2} a c \left (2 b B (1-m)+2 a A \left (\frac {5}{2}-m\right )\right )+\frac {1}{2} c \left (b^2 B (3-2 m)+a (2 A b+a B) (5-2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (A b (5-2 m)+2 a B (3-m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}} \, dx}{c (5-2 m)} \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\text {\$Aborted} \]

[In]

Integrate[(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 2.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \left (a +b \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +\cos \left (f x +e \right ) B \right ) \left (c \sec \left (f x +e \right )\right )^{m}d x\]

[In]

int((a+b*cos(f*x+e))^(3/2)*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^(3/2)*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((B*b*cos(f*x + e)^2 + A*a + (B*a + A*b)*cos(f*x + e))*sqrt(b*cos(f*x + e) + a)*(c*sec(f*x + e))^m, x)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(f*x+e))**(3/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))**m,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 2.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^(3/2)*(c*sec(f*x + e))^m, x)

Giac [N/A]

Not integrable

Time = 1.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^(3/2)*(c*sec(f*x + e))^m, x)

Mupad [N/A]

Not integrable

Time = 5.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2} \,d x \]

[In]

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^(3/2),x)

[Out]

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^(3/2), x)