\(\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\) [173]

Optimal result

Integrand size = 36, antiderivative size = 135 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {2^{\frac {1}{2}+n+\frac {p}{2}} c (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 n-p),\frac {1}{2} (1+2 m+p),\frac {1}{2} (3+2 m+p),\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1}{2} (1-2 n-p)} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{f g (1+2 m+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2932, 2768, 72, 71} \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {c 2^{n+\frac {p}{2}+\frac {1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac {1}{2} (-2 n-p+1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 n-p+1),\frac {1}{2} (2 m+p+1),\frac {1}{2} (2 m+p+3),\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (2 m+p+1)} \]

[In]

Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

[Out]

(2^(1/2 + n + p/2)*c*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 - 2*n - p)/2, (1 + 2*m + p)/2, (3 + 2*m + p
)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 - 2*n - p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-
1 + n))/(f*g*(1 + 2*m + 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 2932

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

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

Mathematica [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx \]

[In]

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

[Out]

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x]

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}d x\]

[In]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

[Out]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

Fricas [F]

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

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**p*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)

Giac [F]

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

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)

Mupad [F(-1)]

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

[In]

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n,x)

[Out]

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x)