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

Optimal. Leaf size=14 \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]

[Out]

(1 + Sin[x]^2)^(3/2)/3

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Rubi [A]  time = 0.0341567, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3198, 261} \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[x]*Sqrt[1 + Sin[x]^2],x]

[Out]

(1 + Sin[x]^2)^(3/2)/3

Rule 3198

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \cos (x) \sin (x) \sqrt{1+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int x \sqrt{1+x^2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{3} \left (1+\sin ^2(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0067136, size = 14, normalized size = 1. \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + Sin[x]^2)^(3/2)/3

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Maple [A]  time = 0.006, size = 11, normalized size = 0.8 \begin{align*}{\frac{1}{3} \left ( 1+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(1+sin(x)^2)^(3/2)

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Maxima [A]  time = 0.964293, size = 14, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (\sin \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sin(x)^2 + 1)^(3/2)

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Fricas [A]  time = 2.14527, size = 36, normalized size = 2.57 \begin{align*} \frac{1}{3} \,{\left (-\cos \left (x\right )^{2} + 2\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-cos(x)^2 + 2)^(3/2)

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Sympy [B]  time = 0.877731, size = 27, normalized size = 1.93 \begin{align*} \frac{\sqrt{\sin ^{2}{\left (x \right )} + 1} \sin ^{2}{\left (x \right )}}{3} + \frac{\sqrt{\sin ^{2}{\left (x \right )} + 1}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(sin(x)**2 + 1)*sin(x)**2/3 + sqrt(sin(x)**2 + 1)/3

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Giac [A]  time = 1.11555, size = 14, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (\sin \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sin(x)^2 + 1)^(3/2)