3.2.25 \(\int (d \sin (e+f x))^n \sqrt {1+\sin (e+f x)} \, dx\) [125]

Optimal. Leaf size=72 \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},1+n;2+n;\sin (e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2855, 66} \begin {gather*} \frac {\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac {1}{2},n+1;n+2;\sin (e+f x)\right )}{d f (n+1) \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 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 {1+\sin (e+f x)} \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {(d 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 {\cos (e+f x) \, _2F_1\left (\frac {1}{2},1+n;2+n;\sin (e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.24, size = 215, normalized size = 2.99 \begin {gather*} \frac {(1-i) 2^{-n} e^{\frac {1}{2} 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 ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt {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*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I)*E^((I/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hyperge
ometric2F1[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))])*(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]])/(2^n*f*(-1 + 2*n)*(
1 + 2*n)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{n} \sqrt {1+\sin \left (f x +e \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral((d*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 \left (d \sin {\left (e + f x \right )}\right )^{n} \sqrt {\sin {\left (e + f x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((d*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 = -0.01 \begin {gather*} \int {\left (d\,\sin \left (e+f\,x\right )\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((d*sin(e + f*x))^n*(sin(e + f*x) + 1)^(1/2),x)

[Out]

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

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