3.718 \(\int \csc ^2(x) (1+\sin ^2(x)) \, dx\)

Optimal. Leaf size=6 \[ x-\cot (x) \]

[Out]

x - Cot[x]

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Rubi [A]  time = 0.0159719, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3012, 8} \[ x-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2*(1 + Sin[x]^2),x]

[Out]

x - Cot[x]

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \csc ^2(x) \left (1+\sin ^2(x)\right ) \, dx &=-\cot (x)+\int 1 \, dx\\ &=x-\cot (x)\\ \end{align*}

Mathematica [A]  time = 0.0028742, size = 6, normalized size = 1. \[ x-\cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2*(1 + Sin[x]^2),x]

[Out]

x - Cot[x]

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Maple [A]  time = 0.017, size = 7, normalized size = 1.2 \begin{align*} x-\cot \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-cot(x)

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Maxima [A]  time = 1.47256, size = 11, normalized size = 1.83 \begin{align*} x - \frac{1}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 1/tan(x)

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Fricas [B]  time = 2.41083, size = 38, normalized size = 6.33 \begin{align*} \frac{x \sin \left (x\right ) - \cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*sin(x) - cos(x))/sin(x)

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Sympy [A]  time = 13.1619, size = 3, normalized size = 0.5 \begin{align*} x - \cot{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - cot(x)

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Giac [B]  time = 1.10361, size = 22, normalized size = 3.67 \begin{align*} x - \frac{1}{2 \, \tan \left (\frac{1}{2} \, x\right )} + \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 1/2/tan(1/2*x) + 1/2*tan(1/2*x)