\(\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx\) [644]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.72 (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 = {} \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx \]

[In]

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

[Out]

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

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))}{(a+b \cos (e+f x))^{3/2}} \, dx \\ & = \frac {2 b (A b-a B) \cos (e+f x) (c \sec (e+f x))^m \sin (e+f x)}{a \left (a^2-b^2\right ) f \sqrt {a+b \cos (e+f x)}}+\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} c \left (a^2 A+A b^2 (1-2 m)-2 a b B (1-m)\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 = 31.59 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((A+B*cos(f*x+e))*(c*sec(f*x+e))^m/(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*sec(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 = 11.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {\left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\left (a + b \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((c*sec(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 = 2.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 8.63 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx=\int \frac {{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[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)