3.5 \(\int \frac {\cot (x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=9 \[ \frac {\log (\sin (x)+1)}{a} \]

[Out]

ln(1+sin(x))/a

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 = {3879, 31} \[ \frac {\log (\sin (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a + a*Csc[x]),x]

[Out]

Log[1 + Sin[x]]/a

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3879

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

Rubi steps

\begin {align*} \int \frac {\cot (x)}{a+a \csc (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\sin (x)\right )\\ &=\frac {\log (1+\sin (x))}{a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {\log (\sin (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a + a*Csc[x]),x]

[Out]

Log[1 + Sin[x]]/a

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fricas [A]  time = 0.70, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*csc(x)),x, algorithm="fricas")

[Out]

log(sin(x) + 1)/a

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giac [A]  time = 0.34, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*csc(x)),x, algorithm="giac")

[Out]

log(sin(x) + 1)/a

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maple [A]  time = 0.32, size = 19, normalized size = 2.11 \[ -\frac {\ln \left (\csc \relax (x )\right )}{a}+\frac {\ln \left (1+\csc \relax (x )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a+a*csc(x)),x)

[Out]

-1/a*ln(csc(x))+1/a*ln(1+csc(x))

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maxima [A]  time = 0.32, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*csc(x)),x, algorithm="maxima")

[Out]

log(sin(x) + 1)/a

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mupad [B]  time = 0.29, size = 25, normalized size = 2.78 \[ \frac {2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + a/sin(x)),x)

[Out]

(2*log(tan(x/2) + 1) - log(tan(x/2)^2 + 1))/a

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot {\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+a*csc(x)),x)

[Out]

Integral(cot(x)/(csc(x) + 1), x)/a

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