3.9 \(\int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=137 \[ -\frac{2 a (A (2 n+3)+2 B (n+1)) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]

[Out]

(-2*a*(2*B*(1 + n) + A*(3 + 2*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*
x])^n)/(f*(3 + 2*n)*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(
d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.213069, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2981, 2776, 67, 65} \[ -\frac{2 a (A (2 n+3)+2 B (n+1)) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sin[e + f*x])^n*Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]),x]

[Out]

(-2*a*(2*B*(1 + n) + A*(3 + 2*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*
x])^n)/(f*(3 + 2*n)*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(
d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2776

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[(a^2*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[(c + d*x)^n/Sqrt[a - b*x]
, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ
[c^2 - d^2, 0] &&  !IntegerQ[2*n]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 65

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

Rubi steps

\begin{align*} \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx &=-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\left (A+\frac{2 B (1+n)}{3+2 n}\right ) \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^2 \left (A+\frac{2 B (1+n)}{3+2 n}\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^2 \left (A+\frac{2 B (1+n)}{3+2 n}\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a \left (A+\frac{2 B (1+n)}{3+2 n}\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 65.8471, size = 409, normalized size = 2.99 \[ -\frac{(1+i) 2^{-n-2} e^{i f n x-\frac{3 i e}{2}} \left (1-e^{2 i (e+f x)}\right )^{-n} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^n \sqrt{a (\sin (e+f x)+1)} \sin ^{-n}(e+f x) (d \sin (e+f x))^n \left (2 e^{i e} \left (\frac{e^{\frac{1}{2} i (2 e+f (1-2 n) x)} \left (i B (2 n-1) e^{i (e+f x)} \, _2F_1\left (\frac{1}{4} (3-2 n),-n;\frac{1}{4} (7-2 n);e^{2 i (e+f x)}\right )-(2 n-3) (2 A+B) \, _2F_1\left (\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right )\right )}{f (2 n-3) (2 n-1)}-\frac{i (2 A+B) e^{-\frac{1}{2} i f (2 n+1) x} \, _2F_1\left (\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right )}{2 f n+f}\right )+\frac{2 B e^{-\frac{1}{2} i f (2 n+3) x} \, _2F_1\left (\frac{1}{4} (-2 n-3),-n;\frac{1}{4} (1-2 n);e^{2 i (e+f x)}\right )}{f (2 n+3)}\right )}{\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sin[e + f*x])^n*Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]),x]

[Out]

((-1 - I)*2^(-2 - n)*E^(((-3*I)/2)*e + I*f*n*x)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^n*((2*B*Hy
pergeometric2F1[(-3 - 2*n)/4, -n, (1 - 2*n)/4, E^((2*I)*(e + f*x))])/(E^((I/2)*f*(3 + 2*n)*x)*f*(3 + 2*n)) + 2
*E^(I*e)*(((-I)*(2*A + B)*Hypergeometric2F1[(-1 - 2*n)/4, -n, (3 - 2*n)/4, E^((2*I)*(e + f*x))])/(E^((I/2)*f*(
1 + 2*n)*x)*(f + 2*f*n)) + (E^((I/2)*(2*e + f*(1 - 2*n)*x))*(-((2*A + B)*(-3 + 2*n)*Hypergeometric2F1[(1 - 2*n
)/4, -n, (5 - 2*n)/4, E^((2*I)*(e + f*x))]) + I*B*E^(I*(e + f*x))*(-1 + 2*n)*Hypergeometric2F1[(3 - 2*n)/4, -n
, (7 - 2*n)/4, E^((2*I)*(e + f*x))]))/(f*(-3 + 2*n)*(-1 + 2*n))))*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])
])/((1 - E^((2*I)*(e + f*x)))^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)

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Maple [F]  time = 0.458, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n}\sqrt{a+a\sin \left ( fx+e \right ) } \left ( A+B\sin \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2)*(A+B*sin(f*x+e)),x)

[Out]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2)*(A+B*sin(f*x+e)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2)*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2)*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*sin(f*x + e) + A)*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (d \sin{\left (e + f x \right )}\right )^{n} \left (A + B \sin{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))**n*(a+a*sin(f*x+e))**(1/2)*(A+B*sin(f*x+e)),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*(d*sin(e + f*x))**n*(A + B*sin(e + f*x)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2)*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: AttributeError