3.87 \(\int \cos (x) (1+\sin ^2(x))^2 \, dx\)

Optimal. Leaf size=19 \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]

[Out]

Sin[x] + (2*Sin[x]^3)/3 + Sin[x]^5/5

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Rubi [A]  time = 0.0191771, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3190, 194} \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*(1 + Sin[x]^2)^2,x]

[Out]

Sin[x] + (2*Sin[x]^3)/3 + Sin[x]^5/5

Rule 3190

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos (x) \left (1+\sin ^2(x)\right )^2 \, dx &=\operatorname{Subst}\left (\int \left (1+x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac{2 \sin ^3(x)}{3}+\frac{\sin ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0035859, size = 19, normalized size = 1. \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*(1 + Sin[x]^2)^2,x]

[Out]

Sin[x] + (2*Sin[x]^3)/3 + Sin[x]^5/5

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Maple [A]  time = 0.01, size = 16, normalized size = 0.8 \begin{align*} \sin \left ( x \right ) +{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)+2/3*sin(x)^3+1/5*sin(x)^5

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Maxima [A]  time = 0.955533, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sin(x)^5 + 2/3*sin(x)^3 + sin(x)

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Fricas [A]  time = 1.63509, size = 61, normalized size = 3.21 \begin{align*} \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 28\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*cos(x)^4 - 16*cos(x)^2 + 28)*sin(x)

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Sympy [A]  time = 1.09388, size = 17, normalized size = 0.89 \begin{align*} \frac{\sin ^{5}{\left (x \right )}}{5} + \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)**5/5 + 2*sin(x)**3/3 + sin(x)

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Giac [A]  time = 1.06927, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sin(x)^5 + 2/3*sin(x)^3 + sin(x)