\(\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx\) [1036]

Optimal result

Integrand size = 38, antiderivative size = 151 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {2^{\frac {1}{2}-\frac {p}{2}} B (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1+p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)} \]

[Out]

(A+B)*(g*cos(f*x+e))^(p+1)*(c-c*sin(f*x+e))^(-1-p)/f/g/(p+1)-2^(1/2-1/2*p)*B*(g*cos(f*x+e))^(p+1)*hypergeom([1
/2+1/2*p, 1/2+1/2*p],[3/2+1/2*p],1/2+1/2*sin(f*x+e))*(1-sin(f*x+e))^(1/2+1/2*p)*(c-c*sin(f*x+e))^(-1-p)/f/g/(p
+1)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2938, 2768, 72, 71} \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B 2^{\frac {1}{2}-\frac {p}{2}} (1-\sin (e+f x))^{\frac {p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {p+1}{2},\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (p+1)} \]

[In]

Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - p),x]

[Out]

((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (2^(1/2 - p/2)*B*(g*Cos[e + f
*x])^(1 + p)*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 +
p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 2768

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[a^2*(
(g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))), Subst[Int[(
a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&
EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]

Rule 2938

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p
+ 1))), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {B \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p} \, dx}{c} \\ & = \frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {\left (B c (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{\frac {1}{2} (-1-p)} (c+c \sin (e+f x))^{\frac {1}{2} (-1-p)}\right ) \text {Subst}\left (\int (c-c x)^{\frac {1}{2} (-1+p)-p} (c+c x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g} \\ & = \frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {\left (2^{-\frac {1}{2}-\frac {p}{2}} B c (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-\frac {1}{2}+\frac {1}{2} (-1-p)-\frac {p}{2}} \left (\frac {c-c \sin (e+f x)}{c}\right )^{\frac {1}{2}+\frac {p}{2}} (c+c \sin (e+f x))^{\frac {1}{2} (-1-p)}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)-p} (c+c x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g} \\ & = \frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {2^{\frac {1}{2}-\frac {p}{2}} B (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1+p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=-\frac {2^{-p/2} \cos (e+f x) (g \cos (e+f x))^p \left (2^{p/2} (A+B)-\sqrt {2} B \operatorname {Hypergeometric2F1}\left (\frac {1+p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}}\right ) (c-c \sin (e+f x))^{-p}}{c f (1+p) (-1+\sin (e+f x))} \]

[In]

Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - p),x]

[Out]

-((Cos[e + f*x]*(g*Cos[e + f*x])^p*(2^(p/2)*(A + B) - Sqrt[2]*B*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p
)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 + p)/2)))/(2^(p/2)*c*f*(1 + p)*(-1 + Sin[e + f*x])*(c - c*Si
n[e + f*x])^p))

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-1-p}d x\]

[In]

int((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-1-p),x)

[Out]

int((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-1-p),x)

Fricas [F]

\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-1-p),x, algorithm="fricas")

[Out]

integral((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^(-p - 1), x)

Sympy [F]

\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- p - 1} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]

[In]

integrate((g*cos(f*x+e))**p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(-1-p),x)

[Out]

Integral((g*cos(e + f*x))**p*(-c*(sin(e + f*x) - 1))**(-p - 1)*(A + B*sin(e + f*x)), x)

Maxima [F]

\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-1-p),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^(-p - 1), x)

Giac [F]

\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-1-p),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^(-p - 1), x)

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+1}} \,d x \]

[In]

int(((g*cos(e + f*x))^p*(A + B*sin(e + f*x)))/(c - c*sin(e + f*x))^(p + 1),x)

[Out]

int(((g*cos(e + f*x))^p*(A + B*sin(e + f*x)))/(c - c*sin(e + f*x))^(p + 1), x)