3.694 \(\int \sec ^2(x) (1+\frac {1}{1+\tan ^2(x)}) \, dx\)

Optimal. Leaf size=4 \[ x+\tan (x) \]

[Out]

x+tan(x)

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Rubi [A]  time = 0.04, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {203} \[ x+\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2*(1 + (1 + Tan[x]^2)^(-1)),x]

[Out]

x + Tan[x]

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])

Rubi steps

\begin {align*} \int \sec ^2(x) \left (1+\frac {1}{1+\tan ^2(x)}\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+\frac {1}{1+x^2}\right ) \, dx,x,\tan (x)\right )\\ &=\tan (x)+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=x+\tan (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ x+\tan (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2*(1 + (1 + Tan[x]^2)^(-1)),x]

[Out]

x + Tan[x]

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fricas [B]  time = 0.89, size = 12, normalized size = 3.00 \[ \frac {x \cos \relax (x) + \sin \relax (x)}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(1+1/(1+tan(x)^2)),x, algorithm="fricas")

[Out]

(x*cos(x) + sin(x))/cos(x)

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giac [A]  time = 0.13, size = 4, normalized size = 1.00 \[ x + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(1+1/(1+tan(x)^2)),x, algorithm="giac")

[Out]

x + tan(x)

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maple [A]  time = 0.12, size = 5, normalized size = 1.25 \[ x +\tan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*(1+1/(1+tan(x)^2)),x)

[Out]

x+tan(x)

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maxima [A]  time = 0.41, size = 4, normalized size = 1.00 \[ x + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(1+1/(1+tan(x)^2)),x, algorithm="maxima")

[Out]

x + tan(x)

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mupad [B]  time = 2.94, size = 4, normalized size = 1.00 \[ x+\mathrm {tan}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(tan(x)^2 + 1) + 1)/cos(x)^2,x)

[Out]

x + tan(x)

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sympy [B]  time = 0.74, size = 27, normalized size = 6.75 \[ \frac {x \sec ^{2}{\relax (x )}}{\tan ^{2}{\relax (x )} + 1} + \frac {\tan {\relax (x )} \sec ^{2}{\relax (x )}}{\tan ^{2}{\relax (x )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*(1+1/(1+tan(x)**2)),x)

[Out]

x*sec(x)**2/(tan(x)**2 + 1) + tan(x)*sec(x)**2/(tan(x)**2 + 1)

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