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

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

[Out]

Log[1 + Sin[x]]/a

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Rubi [A]  time = 0.0217797, antiderivative size = 9, 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 = {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 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]

Rule 31

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

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*}

Mathematica [A]  time = 0.0085864, size = 9, normalized size = 1. \[ \frac{\log (\sin (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a + a*Csc[x]),x]

[Out]

Log[1 + Sin[x]]/a

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Maple [A]  time = 0.048, size = 19, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+\csc \left ( x \right ) \right ) }{a}}-{\frac{\ln \left ( \csc \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.980983, size = 12, normalized size = 1.33 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{a} \end{align*}

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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Fricas [A]  time = 0.480226, size = 26, normalized size = 2.89 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{a} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

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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Giac [A]  time = 1.29816, size = 12, normalized size = 1.33 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{a} \end{align*}

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