\(\int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx\) [338]

Optimal result

Integrand size = 25, antiderivative size = 97 \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2^{1+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d e (1+p) \sqrt {a+a \sin (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71} \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {a 2^{\frac {p}{2}+1} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1) \sqrt {a \sin (c+d x)+a}} \]

[In]

Int[(e*Cos[c + d*x])^p*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-((2^(1 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[-1/2*p, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2
])/(d*e*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a + a*Sin[c + d*x]]))

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]

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2^{1+\frac {p}{2}} a \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d (1+p) \sqrt {a (1+\sin (c+d x))}} \]

[In]

Integrate[(e*Cos[c + d*x])^p*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-((2^(1 + p/2)*a*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[-1/2*p, (1 + p)/2, (3 + p)/2, (1 - Sin[c +
d*x])/2])/(d*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a*(1 + Sin[c + d*x])]))

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{p} \sqrt {a +a \sin \left (d x +c \right )}d x\]

[In]

int((e*cos(d*x+c))^p*(a+a*sin(d*x+c))^(1/2),x)

[Out]

int((e*cos(d*x+c))^p*(a+a*sin(d*x+c))^(1/2),x)

Fricas [F]

\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

integrate((e*cos(d*x+c))^p*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Sympy [F]

\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{p}\, dx \]

[In]

integrate((e*cos(d*x+c))**p*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(c + d*x) + 1))*(e*cos(c + d*x))**p, x)

Maxima [F]

\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

integrate((e*cos(d*x+c))^p*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Giac [F]

\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

integrate((e*cos(d*x+c))^p*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Mupad [F(-1)]

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

[In]

int((e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(1/2),x)

[Out]

int((e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(1/2), x)