\(\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1019]

Optimal result

Integrand size = 33, antiderivative size = 170 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {2^{\frac {1}{2} (1+2 m+p)} a (B m+A (1+m+p)) (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 m-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2} (1-2 m-p)} (a+a \sin (e+f x))^{-1+m}}{f g (1+p) (1+m+p)}-\frac {B (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g (1+m+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2939, 2768, 72, 71} \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {a 2^{\frac {1}{2} (2 m+p+1)} (A (m+p+1)+B m) (a \sin (e+f x)+a)^{m-1} (g \cos (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac {1}{2} (-2 m-p+1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 m-p+1),\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f g (p+1) (m+p+1)}-\frac {B (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g (m+p+1)} \]

[In]

Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

-((2^((1 + 2*m + p)/2)*a*(B*m + A*(1 + m + p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 - 2*m - p)/2, (1
+ p)/2, (3 + p)/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^((1 - 2*m - p)/2)*(a + a*Sin[e + f*x])^(-1 + m))/(
f*g*(1 + p)*(1 + m + p))) - (B*(g*Cos[e + f*x])^(1 + p)*(a + a*Sin[e + f*x])^m)/(f*g*(1 + 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 2939

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

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {\cos (e+f x) (g \cos (e+f x))^p (1+\sin (e+f x))^{\frac {1}{2} (-1-2 m-p)} (a (1+\sin (e+f x)))^m \left (2^{\frac {1}{2} (1+2 m+p)} (B m+A (1+m+p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 m-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (e+f x))\right )+B (1+p) (1+\sin (e+f x))^{\frac {1}{2} (1+2 m+p)}\right )}{f (1+p) (1+m+p)} \]

[In]

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

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

Maple [F]

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

[In]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

[Out]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

Fricas [F]

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

[In]

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

[Out]

integral((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m, x)

Sympy [F]

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

[In]

integrate((g*cos(f*x+e))**p*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m, x)

Mupad [F(-1)]

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

[In]

int((g*cos(e + f*x))^p*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m,x)

[Out]

int((g*cos(e + f*x))^p*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m, x)