\(\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx\) [458]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.53 (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 = {} \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 32.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.90 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {\left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\left (a + b \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {{\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[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)