3.114 \(\int \sin (x) \sqrt{1+\sin ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ -\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right ) \]

[Out]

-ArcSin[Cos[x]/Sqrt[2]] - (Cos[x]*Sqrt[2 - Cos[x]^2])/2

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Rubi [A]  time = 0.02998, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3186, 195, 216} \[ -\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]*Sqrt[1 + Sin[x]^2],x]

[Out]

-ArcSin[Cos[x]/Sqrt[2]] - (Cos[x]*Sqrt[2 - Cos[x]^2])/2

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sin (x) \sqrt{1+\sin ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \sqrt{2-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2}} \, dx,x,\cos (x)\right )\\ &=-\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right )-\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}\\ \end{align*}

Mathematica [C]  time = 0.048358, size = 53, normalized size = 1.77 \[ -\frac{\cos (x) \sqrt{3-\cos (2 x)}}{2 \sqrt{2}}+i \log \left (\sqrt{3-\cos (2 x)}+i \sqrt{2} \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]*Sqrt[1 + Sin[x]^2],x]

[Out]

-(Cos[x]*Sqrt[3 - Cos[2*x]])/(2*Sqrt[2]) + I*Log[I*Sqrt[2]*Cos[x] + Sqrt[3 - Cos[2*x]]]

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Maple [A]  time = 0.938, size = 51, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,\cos \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \sqrt{- \left ( \cos \left ( x \right ) \right ) ^{4}+2\, \left ( \cos \left ( x \right ) \right ) ^{2}}+\arcsin \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) \right ){\frac{1}{\sqrt{1+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((1+sin(x)^2)*cos(x)^2)^(1/2)*((-cos(x)^4+2*cos(x)^2)^(1/2)+arcsin(cos(x)^2-1))/cos(x)/(1+sin(x)^2)^(1/2)

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Maxima [A]  time = 1.41041, size = 34, normalized size = 1.13 \begin{align*} -\frac{1}{2} \, \sqrt{-\cos \left (x\right )^{2} + 2} \cos \left (x\right ) - \arcsin \left (\frac{1}{2} \, \sqrt{2} \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(1+sin(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-cos(x)^2 + 2)*cos(x) - arcsin(1/2*sqrt(2)*cos(x))

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Fricas [B]  time = 1.79444, size = 219, normalized size = 7.3 \begin{align*} -\frac{1}{2} \, \sqrt{-\cos \left (x\right )^{2} + 2} \cos \left (x\right ) + \frac{1}{2} \, \arctan \left (-\frac{\cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt{-\cos \left (x\right )^{2} + 2}}{\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 1}\right ) - \frac{1}{2} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-cos(x)^2 + 2)*cos(x) + 1/2*arctan(-(cos(x)*sin(x) - (cos(x)^3 - cos(x))*sqrt(-cos(x)^2 + 2))/(cos(x
)^4 - 3*cos(x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin ^{2}{\left (x \right )} + 1} \sin{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(1+sin(x)**2)**(1/2),x)

[Out]

Integral(sqrt(sin(x)**2 + 1)*sin(x), x)

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Giac [A]  time = 1.19713, size = 34, normalized size = 1.13 \begin{align*} -\frac{1}{2} \, \sqrt{-\cos \left (x\right )^{2} + 2} \cos \left (x\right ) - \arcsin \left (\frac{1}{2} \, \sqrt{2} \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(1+sin(x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-cos(x)^2 + 2)*cos(x) - arcsin(1/2*sqrt(2)*cos(x))