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

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 = 0.033788, 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 \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]

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

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Rubi in Sympy [A]  time = 2.29008, size = 17, normalized size = 0.89 \[ \frac{\sin ^{5}{\left (x \right )}}{5} + \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.0055981, 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 = 0.009, size = 16, normalized size = 0.8 \[ \sin \left ( x \right ) +{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}} \]

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 = 1.38825, size = 20, normalized size = 1.05 \[ \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sin(x)^5 + 2/3*sin(x)^3 + sin(x)

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Fricas [A]  time = 0.22883, size = 24, normalized size = 1.26 \[ \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 28\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.58961, size = 17, normalized size = 0.89 \[ \frac{\sin ^{5}{\left (x \right )}}{5} + \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

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/XCAS [A]  time = 0.205921, size = 20, normalized size = 1.05 \[ \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sin(x)^5 + 2/3*sin(x)^3 + sin(x)