Integrand size = 38, antiderivative size = 151 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {2^{\frac {1}{2}-\frac {p}{2}} B (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1+p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)} \]
(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 )
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=-\frac {2^{-p/2} \cos (e+f x) (g \cos (e+f x))^p \left (2^{p/2} (A+B)-\sqrt {2} B \operatorname {Hypergeometric2F1}\left (\frac {1+p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}}\right ) (c-c \sin (e+f x))^{-p}}{c f (1+p) (-1+\sin (e+f x))} \]
-((Cos[e + f*x]*(g*Cos[e + f*x])^p*(2^(p/2)*(A + B) - Sqrt[2]*B*Hypergeome tric2F1[(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*Sin[ e + f*x])^p))
Time = 0.53 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3338, 3042, 3168, 80, 79}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^pdx\) |
\(\Big \downarrow \) 3338 |
\(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p}dx}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p}dx}{c}\) |
\(\Big \downarrow \) 3168 |
\(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B c (c-c \sin (e+f x))^{\frac {1}{2} (-p-1)} (c \sin (e+f x)+c)^{\frac {1}{2} (-p-1)} (g \cos (e+f x))^{p+1} \int (c-c \sin (e+f x))^{\frac {1}{2} (-p-1)} (\sin (e+f x) c+c)^{\frac {p-1}{2}}d\sin (e+f x)}{f g}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B c 2^{-\frac {p}{2}-\frac {1}{2}} (1-\sin (e+f x))^{\frac {p+1}{2}} (c-c \sin (e+f x))^{-p-1} (c \sin (e+f x)+c)^{\frac {1}{2} (-p-1)} (g \cos (e+f x))^{p+1} \int \left (\frac {1}{2}-\frac {1}{2} \sin (e+f x)\right )^{\frac {1}{2} (-p-1)} (\sin (e+f x) c+c)^{\frac {p-1}{2}}d\sin (e+f x)}{f g}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac {B 2^{\frac {1}{2}-\frac {p}{2}} (1-\sin (e+f x))^{\frac {p+1}{2}} (c-c \sin (e+f x))^{-p-1} (c \sin (e+f x)+c)^{\frac {1}{2} (-p-1)+\frac {p+1}{2}} (g \cos (e+f x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {p+1}{2},\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (p+1)}\) |
((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)*(c + c*Sin[e + f*x])^((-1 - p)/2 + (1 + p)/2))/(f*g*(1 + p))
3.11.36.3.1 Defintions of rubi rules used
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] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(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] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[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] /; Fre eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
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] + Simp[(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]
\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-1-p}d x\]
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- p - 1} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 1} \,d x } \]
Timed out. \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+1}} \,d x \]