3.2.29 \(\int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx\) [129]

Optimal. Leaf size=66 \[ -\frac {2 a \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2855, 69, 67} \begin {gather*} -\frac {2 a \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 \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 67

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

Rule 69

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 2855

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]

Rubi steps

\begin {align*} \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx &=\frac {\left (a^2 \cos (e+f x)\right ) \text {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 {\left (a^2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {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 \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)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 266, normalized size = 4.03 \begin {gather*} \frac {(1+i) e^{-\frac {1}{2} i f x} \left (e^{i f x} (1+2 n) \, _2F_1\left (\frac {1}{4} (1-2 n),-n;\frac {1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right ) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right )+(-1+2 n) \, _2F_1\left (\frac {1}{4} (-1-2 n),-n;\frac {1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right ) \left (i \cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )\right ) \left (1-e^{2 i f x} \cos ^2(e)+e^{2 i f x} \sin ^2(e)-i e^{2 i f x} \sin (2 e)\right )^{-n} (d \sin (e+f x))^n \sqrt {a (1+\sin (e+f x))}}{f (-1+2 n) (1+2 n) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{n} \sqrt {a +a \sin \left (f x +e \right )}\, dx\]

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),x)

[Out]

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

[Out]

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

[Out]

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

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

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),x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\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((d*sin(e + f*x))^n*(a + a*sin(e + f*x))^(1/2),x)

[Out]

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

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