3.3.46 \(\int (\frac {1}{1+\cos (x)}+\frac {1}{1+\cot (x)}+\frac {1}{1+\csc (x)}+\frac {1}{1+\sec (x)}+\frac {1}{1+\sin (x)}+\frac {1}{1+\tan (x)}) \, dx\) [246]

Optimal. Leaf size=3 \[ 3 x \]

[Out]

3*x

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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
time = 0.06, antiderivative size = 42, normalized size of antiderivative = 14.00, number of steps used = 11, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2727, 3565, 3611, 3862, 8} \begin {gather*} 3 x+\frac {\sin (x)}{\cos (x)+1}-\frac {\cos (x)}{\sin (x)+1}+\frac {\cot (x)}{\csc (x)+1}-\frac {\tan (x)}{\sec (x)+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x])^(-1) + (1 + Cot[x])^(-1) + (1 + Csc[x])^(-1) + (1 + Sec[x])^(-1) + (1 + Sin[x])^(-1) + (1 + T
an[x])^(-1),x]

[Out]

3*x + Cot[x]/(1 + Csc[x]) + Sin[x]/(1 + Cos[x]) - Cos[x]/(1 + Sin[x]) - Tan[x]/(1 + Sec[x])

Rule 8

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

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3565

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[a*(x/(a^2 + b^2)), x] + Dist[b/(a^2 + b^2),
Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3862

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-Cot[c + d*x])*((a + b*Csc[c + d*x])^n/(d*
(2*n + 1))), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*
x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {1}{1+\cos (x)} \, dx+\int \frac {1}{1+\cot (x)} \, dx+\int \frac {1}{1+\csc (x)} \, dx+\int \frac {1}{1+\sec (x)} \, dx+\int \frac {1}{1+\sin (x)} \, dx+\int \frac {1}{1+\tan (x)} \, dx\\ &=x+\frac {\cot (x)}{1+\csc (x)}+\frac {\sin (x)}{1+\cos (x)}-\frac {\cos (x)}{1+\sin (x)}-\frac {\tan (x)}{1+\sec (x)}-\frac {1}{2} \int \frac {-1+\cot (x)}{1+\cot (x)} \, dx+\frac {1}{2} \int \frac {1-\tan (x)}{1+\tan (x)} \, dx-2 \int -1 \, dx\\ &=3 x+\frac {\cot (x)}{1+\csc (x)}+\frac {\sin (x)}{1+\cos (x)}-\frac {\cos (x)}{1+\sin (x)}-\frac {\tan (x)}{1+\sec (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} 3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x])^(-1) + (1 + Cot[x])^(-1) + (1 + Csc[x])^(-1) + (1 + Sec[x])^(-1) + (1 + Sin[x])^(-1) +
(1 + Tan[x])^(-1),x]

[Out]

3*x

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.39, size = 54, normalized size = 18.00

method	result	size
risch	\(3 x\)	\(4\)
norman	\(\frac {3 x +3 x \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )}\)	\(21\)
default	\(\frac {\ln \left (1+\cot ^{2}\left (x \right )\right )}{4}-\frac {\pi }{4}+\frac {\mathrm {arccot}\left (\cot \left (x \right )\right )}{2}-\frac {\ln \left (1+\cot \left (x \right )\right )}{2}+4 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{4}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{2}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(1+cot(x)^2)-1/4*Pi+1/2*arccot(cot(x))-1/2*ln(1+cot(x))+4*arctan(tan(1/2*x))-1/4*ln(1+tan(x)^2)+1/2*arct
an(tan(x))+1/2*ln(1+tan(x))

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Maxima [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, size = 14, normalized size = 4.67 \begin {gather*} x + 4 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x, algorithm="maxima")

[Out]

x + 4*arctan(sin(x)/(cos(x) + 1))

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Fricas [A]
time = 0.56, size = 3, normalized size = 1.00 \begin {gather*} 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x, algorithm="fricas")

[Out]

3*x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} + \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (x \right )} + \sin {\left (x \right )} \cos {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \cos {\left (x \right )} \cot {\left (x \right )} + 2 \sin {\left (x \right )} \cos {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \cos {\left (x \right )} \csc {\left (x \right )} + 3 \sin {\left (x \right )} \cos {\left (x \right )} \sec {\left (x \right )} + 4 \sin {\left (x \right )} \cos {\left (x \right )} + \sin {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 2 \sin {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 2 \sin {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} + 2 \sin {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} + 3 \sin {\left (x \right )} \tan {\left (x \right )} \sec {\left (x \right )} + 4 \sin {\left (x \right )} \tan {\left (x \right )} + 2 \sin {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \sin {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 3 \sin {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 4 \sin {\left (x \right )} \cot {\left (x \right )} + 3 \sin {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 4 \sin {\left (x \right )} \csc {\left (x \right )} + 4 \sin {\left (x \right )} \sec {\left (x \right )} + 5 \sin {\left (x \right )} + \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 2 \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 2 \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 3 \cos {\left (x \right )} \tan {\left (x \right )} \cot {\left (x \right )} + 2 \cos {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \cos {\left (x \right )} \tan {\left (x \right )} \csc {\left (x \right )} + 3 \cos {\left (x \right )} \tan {\left (x \right )} \sec {\left (x \right )} + 4 \cos {\left (x \right )} \tan {\left (x \right )} + 2 \cos {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \cos {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 3 \cos {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 4 \cos {\left (x \right )} \cot {\left (x \right )} + 3 \cos {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 4 \cos {\left (x \right )} \csc {\left (x \right )} + 4 \cos {\left (x \right )} \sec {\left (x \right )} + 5 \cos {\left (x \right )} + 2 \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 3 \tan {\left (x \right )} \cot {\left (x \right )} \csc {\left (x \right )} + 3 \tan {\left (x \right )} \cot {\left (x \right )} \sec {\left (x \right )} + 4 \tan {\left (x \right )} \cot {\left (x \right )} + 3 \tan {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 4 \tan {\left (x \right )} \csc {\left (x \right )} + 4 \tan {\left (x \right )} \sec {\left (x \right )} + 5 \tan {\left (x \right )} + 3 \cot {\left (x \right )} \csc {\left (x \right )} \sec {\left (x \right )} + 4 \cot {\left (x \right )} \csc {\left (x \right )} + 4 \cot {\left (x \right )} \sec {\left (x \right )} + 5 \cot {\left (x \right )} + 4 \csc {\left (x \right )} \sec {\left (x \right )} + 5 \csc {\left (x \right )} + 5 \sec {\left (x \right )} + 6}{\left (\sin {\left (x \right )} + 1\right ) \left (\cos {\left (x \right )} + 1\right ) \left (\tan {\left (x \right )} + 1\right ) \left (\cot {\left (x \right )} + 1\right ) \left (\csc {\left (x \right )} + 1\right ) \left (\sec {\left (x \right )} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x)

[Out]

Integral((sin(x)*cos(x)*tan(x)*cot(x)*csc(x) + sin(x)*cos(x)*tan(x)*cot(x)*sec(x) + 2*sin(x)*cos(x)*tan(x)*cot
(x) + sin(x)*cos(x)*tan(x)*csc(x)*sec(x) + 2*sin(x)*cos(x)*tan(x)*csc(x) + 2*sin(x)*cos(x)*tan(x)*sec(x) + 3*s
in(x)*cos(x)*tan(x) + sin(x)*cos(x)*cot(x)*csc(x)*sec(x) + 2*sin(x)*cos(x)*cot(x)*csc(x) + 2*sin(x)*cos(x)*cot
(x)*sec(x) + 3*sin(x)*cos(x)*cot(x) + 2*sin(x)*cos(x)*csc(x)*sec(x) + 3*sin(x)*cos(x)*csc(x) + 3*sin(x)*cos(x)
*sec(x) + 4*sin(x)*cos(x) + sin(x)*tan(x)*cot(x)*csc(x)*sec(x) + 2*sin(x)*tan(x)*cot(x)*csc(x) + 2*sin(x)*tan(
x)*cot(x)*sec(x) + 3*sin(x)*tan(x)*cot(x) + 2*sin(x)*tan(x)*csc(x)*sec(x) + 3*sin(x)*tan(x)*csc(x) + 3*sin(x)*
tan(x)*sec(x) + 4*sin(x)*tan(x) + 2*sin(x)*cot(x)*csc(x)*sec(x) + 3*sin(x)*cot(x)*csc(x) + 3*sin(x)*cot(x)*sec
(x) + 4*sin(x)*cot(x) + 3*sin(x)*csc(x)*sec(x) + 4*sin(x)*csc(x) + 4*sin(x)*sec(x) + 5*sin(x) + cos(x)*tan(x)*
cot(x)*csc(x)*sec(x) + 2*cos(x)*tan(x)*cot(x)*csc(x) + 2*cos(x)*tan(x)*cot(x)*sec(x) + 3*cos(x)*tan(x)*cot(x)
+ 2*cos(x)*tan(x)*csc(x)*sec(x) + 3*cos(x)*tan(x)*csc(x) + 3*cos(x)*tan(x)*sec(x) + 4*cos(x)*tan(x) + 2*cos(x)
*cot(x)*csc(x)*sec(x) + 3*cos(x)*cot(x)*csc(x) + 3*cos(x)*cot(x)*sec(x) + 4*cos(x)*cot(x) + 3*cos(x)*csc(x)*se
c(x) + 4*cos(x)*csc(x) + 4*cos(x)*sec(x) + 5*cos(x) + 2*tan(x)*cot(x)*csc(x)*sec(x) + 3*tan(x)*cot(x)*csc(x) +
3*tan(x)*cot(x)*sec(x) + 4*tan(x)*cot(x) + 3*tan(x)*csc(x)*sec(x) + 4*tan(x)*csc(x) + 4*tan(x)*sec(x) + 5*tan
(x) + 3*cot(x)*csc(x)*sec(x) + 4*cot(x)*csc(x) + 4*cot(x)*sec(x) + 5*cot(x) + 4*csc(x)*sec(x) + 5*csc(x) + 5*s
ec(x) + 6)/((sin(x) + 1)*(cos(x) + 1)*(tan(x) + 1)*(cot(x) + 1)*(csc(x) + 1)*(sec(x) + 1)), x)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.50, size = 40, normalized size = 13.33 \begin {gather*} 3 \, x - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (x^{2} + 1\right )} {\left (\frac {x^{2} - 1}{x^{2} + 1} - 1\right )}} - \tan \left (\frac {1}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x, algorithm="giac")

[Out]

3*x - 2*tan(1/2*x)/((x^2 + 1)*((x^2 - 1)/(x^2 + 1) - 1)) - tan(1/2*x)

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Mupad [B]
time = 0.36, size = 3, normalized size = 1.00 \begin {gather*} 3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/sin(x) + 1) + 1/(cos(x) + 1) + 1/(cot(x) + 1) + 1/(sin(x) + 1) + 1/(tan(x) + 1) + 1/(1/cos(x) + 1),x)

[Out]

3*x

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Chatgpt [F] Failed to verify
time = 1.00, size = 13, normalized size = 4.33 \begin {gather*} 3 \ln \left (2 \left (\sin ^{2}\left (x +\frac {\pi }{4}\right )\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/(1+sin(x))+1/(1+cos(x))+1/(1+tan(x))+1/(1+cot(x))+1/(1+sec(x))+1/(1+csc(x)),x)

[Out]

3*ln(2*sin(x+1/4*Pi)^2)

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