3.5.18 \(\int \frac {(c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx\) [418]

Optimal. Leaf size=68 \[ \frac {\cos (e+f x) \, _2F_1\left (1,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt {a-a \sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2824, 2746, 70} \begin {gather*} \frac {\cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left (1,m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt {a-a \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2824

Int[((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]/Cos[e + f*x]^(2*
FracPart[m])), Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] ||  !FractionQ[n])

Rubi steps

\begin {align*} \int \frac {(c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx &=\frac {\cos (e+f x) \int \sec (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 {(c \cos (e+f x)) \text {Subst}\left (\int \frac {(c+x)^{-\frac {1}{2}+m}}{c-x} \, dx,x,c \sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) \, _2F_1\left (1,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt {a-a \sin (e+f x)}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(157\) vs. \(2(68)=136\).
time = 0.24, size = 157, normalized size = 2.31 \begin {gather*} \frac {2^{-\frac {3}{2}-2 m} \left (4^m \, _2F_1\left (1,2 m;1+2 m;\sin \left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right )-\, _2F_1\left (2 m,2 m;1+2 m;\frac {1}{2} \left (1-\tan ^2\left (\frac {1}{8} (2 e-\pi +2 f x)\right )\right )\right ) \sec ^2\left (\frac {1}{8} (2 e-\pi +2 f x)\right )^{2 m}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c (1+\sin (e+f x)))^m}{f m \sqrt {a-a \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-3/2 - 2*m)*(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Sin[(2*e + Pi + 2*f*x)/4]] - Hypergeometric2F1[2*m, 2*
m, 1 + 2*m, (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^m)/(f*m*Sqrt[a - a*Sin[e + f*x]])

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (c +c \sin \left (f x +e \right )\right )^{m}}{\sqrt {a -a \sin \left (f x +e \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*(sin(e + f*x) + 1))**m/sqrt(-a*(sin(e + f*x) - 1)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+c\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {a-a\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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