3.2.17 \(\int \sin ^n(e+f x) \sqrt {1+\sin (e+f x)} \, dx\) [117]

Optimal. Leaf size=43 \[ -\frac {2 \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {1+\sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2855, 67} \begin {gather*} -\frac {2 \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {\sin (e+f x)+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]

[Out]

(-2*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[1 + 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 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 \sin ^n(e+f x) \sqrt {1+\sin (e+f x)} \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {x^n}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {2 \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {1+\sin (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.45, size = 186, normalized size = 4.33 \begin {gather*} \frac {2^{1-n} e^{i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{1+n} \left (i (-1+2 n) \, _2F_1\left (1,\frac {1}{4} (3+2 n);\frac {1}{4} (3-2 n);e^{2 i (e+f x)}\right )+e^{i (e+f x)} (1+2 n) \, _2F_1\left (1,\frac {1}{4} (5+2 n);\frac {1}{4} (5-2 n);e^{2 i (e+f x)}\right )\right ) \sqrt {1+\sin (e+f x)}}{\left (i+e^{i (e+f x)}\right ) f (-1+2 n) (1+2 n)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]

[Out]

(2^(1 - n)*E^(I*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hypergeom
etric2F1[1, (3 + 2*n)/4, (3 - 2*n)/4, E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(1 + 2*n)*Hypergeometric2F1[1, (5
+ 2*n)/4, (5 - 2*n)/4, E^((2*I)*(e + f*x))])*Sqrt[1 + Sin[e + f*x]])/((I + E^(I*(e + f*x)))*f*(-1 + 2*n)*(1 +
2*n))

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (\sin ^{n}\left (f x +e \right )\right ) \sqrt {1+\sin \left (f x +e \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**n*(1+sin(f*x+e))**(1/2),x)

[Out]

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

[Out]

integrate(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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