\(\int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx\) [166]

Optimal result

Integrand size = 40, antiderivative size = 106 \[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=-\frac {2^{\frac {9}{4}+m} c \cos (e+f x) (g \cos (e+f x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-\frac {1}{4}-m,\frac {7}{4},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (c+c \sin (e+f x))^{-1+m}}{3 f \sqrt {a-a \sin (e+f x)}} \]

[Out]

-1/3*2^(9/4+m)*c*cos(f*x+e)*(g*cos(f*x+e))^(3/2)*hypergeom([3/4, -1/4-m],[7/4],1/2-1/2*sin(f*x+e))*(1+sin(f*x+
e))^(-1/4-m)*(c+c*sin(f*x+e))^(-1+m)/f/(a-a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2932, 2768, 72, 71} \[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=-\frac {c 2^{m+\frac {9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (c \sin (e+f x)+c)^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-m-\frac {1}{4},\frac {7}{4},\frac {1}{2} (1-\sin (e+f x))\right )}{3 f \sqrt {a-a \sin (e+f x)}} \]

[In]

Int[((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]

[Out]

-1/3*(2^(9/4 + m)*c*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -1/4 - m, 7/4, (1 - Sin[e + f*x
])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(c + c*Sin[e + f*x])^(-1 + m))/(f*Sqrt[a - a*Sin[e + f*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]

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}& = \frac {(g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} (c+c \sin (e+f x))^{\frac {1}{2}+m} \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {\left (c^2 \cos (e+f x) (g \cos (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {(c+c x)^{\frac {1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4} (c+c \sin (e+f x))^{5/4}} \\ & = \frac {\left (2^{\frac {1}{4}+m} c^2 \cos (e+f x) (g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^{-1+m} \left (\frac {c+c \sin (e+f x)}{c}\right )^{-\frac {1}{4}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4}} \\ & = -\frac {2^{\frac {9}{4}+m} c \cos (e+f x) (g \cos (e+f x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-\frac {1}{4}-m,\frac {7}{4},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (c+c \sin (e+f x))^{-1+m}}{3 f \sqrt {a-a \sin (e+f x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]

[Out]

Integrate[((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]], x]

Maple [F]

\[\int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +c \sin \left (f x +e \right )\right )^{m}}{\sqrt {a -a \sin \left (f x +e \right )}}d x\]

[In]

int((g*cos(f*x+e))^(3/2)*(c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x)

[Out]

int((g*cos(f*x+e))^(3/2)*(c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x)

Fricas [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(c+c*sin(f*x+e))**m/(a-a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(c*sin(f*x + e) + c)^m/sqrt(-a*sin(f*x + e) + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+c\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {a-a\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int(((g*cos(e + f*x))^(3/2)*(c + c*sin(e + f*x))^m)/(a - a*sin(e + f*x))^(1/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(c + c*sin(e + f*x))^m)/(a - a*sin(e + f*x))^(1/2), x)