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

Optimal. Leaf size=209 \[ \frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Unintegrable}\left (\frac{(c \cos (e+f x))^{-m} \left (\frac{1}{2} c \cos (e+f x) \left (a (5-2 m) (a B+2 A b)+b^2 B (3-2 m)\right )+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (3-m)+A b (5-2 m))+\frac{1}{2} a c \left (2 a A \left (\frac{5}{2}-m\right )+2 b B (1-m)\right )\right )}{\sqrt{a+b \cos (e+f x)}},x\right )}{c (5-2 m)}+\frac{2 b B \sin (e+f x) \cos (e+f x) \sqrt{a+b \cos (e+f x)} (c \sec (e+f x))^m}{f (5-2 m)} \]

[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*Unintegrable[((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 [A]  time = 0.721117, 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 = {} \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[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*} \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=\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 [A]  time = 57.0767, size = 0, normalized size = 0. \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.714, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b \cos \left (f x + e\right )^{2} + A a +{\left (B a + A b\right )} \cos \left (f x + e\right )\right )} \sqrt{b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)