3.7 \(\int \frac {\csc ^2(2+3 x)}{2+\cot ^2(2+3 x)} \, dx\)

Optimal. Leaf size=48 \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]

[Out]

1/2*x*2^(1/2)+1/6*arctan(cos(2+3*x)*sin(2+3*x)/(1+sin(2+3*x)^2+2^(1/2)))*2^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3675, 203} \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[2 + 3*x]^2/(2 + Cot[2 + 3*x]^2),x]

[Out]

x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])

Rule 203

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

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 22, normalized size = 0.46 \[ \frac {\tan ^{-1}\left (\sqrt {2} \tan (3 x+2)\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[2 + 3*x]^2/(2 + Cot[2 + 3*x]^2),x]

[Out]

ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])

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fricas [A]  time = 0.64, size = 43, normalized size = 0.90 \[ -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (3 \, x + 2\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (3 \, x + 2\right ) \sin \left (3 \, x + 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2+3*x)^2/(2+cot(2+3*x)^2),x, algorithm="fricas")

[Out]

-1/12*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(3*x + 2)^2 - 2*sqrt(2))/(cos(3*x + 2)*sin(3*x + 2)))

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giac [A]  time = 0.28, size = 57, normalized size = 1.19 \[ \frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - 2 \, \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - 2 \, \cos \left (6 \, x + 4\right ) + 2}\right ) + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2+3*x)^2/(2+cot(2+3*x)^2),x, algorithm="giac")

[Out]

1/6*sqrt(2)*(3*x + arctan(-(sqrt(2)*sin(6*x + 4) - 2*sin(6*x + 4))/(sqrt(2)*cos(6*x + 4) + sqrt(2) - 2*cos(6*x
+ 4) + 2)) + 2)

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maple [A]  time = 0.29, size = 17, normalized size = 0.35 \[ \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, \tan \left (2+3 x \right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2+3*x)^2/(2+cot(2+3*x)^2),x)

[Out]

1/6*2^(1/2)*arctan(2^(1/2)*tan(2+3*x))

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maxima [A]  time = 0.61, size = 16, normalized size = 0.33 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \left (3 \, x + 2\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2+3*x)^2/(2+cot(2+3*x)^2),x, algorithm="maxima")

[Out]

1/6*sqrt(2)*arctan(sqrt(2)*tan(3*x + 2))

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mupad [B]  time = 2.40, size = 16, normalized size = 0.33 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(3*x + 2)^2*(cot(3*x + 2)^2 + 2)),x)

[Out]

(2^(1/2)*atan(2^(1/2)*tan(3*x + 2)))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (3 x + 2 \right )}}{\cot ^{2}{\left (3 x + 2 \right )} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2+3*x)**2/(2+cot(2+3*x)**2),x)

[Out]

Integral(csc(3*x + 2)**2/(cot(3*x + 2)**2 + 2), x)

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