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

Optimal. Leaf size=19 \[ \frac {\log (a+b \csc (x))}{a}+\frac {\log (\sin (x))}{a} \]

[Out]

ln(a+b*csc(x))/a+ln(sin(x))/a

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac {\log (a+b \csc (x))}{a}+\frac {\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a + b*Csc[x]),x]

[Out]

Log[a + b*Csc[x]]/a + Log[Sin[x]]/a

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

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

Rule 3885

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.58 \[ \frac {\log (a \sin (x)+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a + b*Csc[x]),x]

[Out]

Log[b + a*Sin[x]]/a

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fricas [A]  time = 0.55, size = 11, normalized size = 0.58 \[ \frac {\log \left (a \sin \relax (x) + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*sin(x) + b)/a

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giac [A]  time = 0.42, size = 12, normalized size = 0.63 \[ \frac {\log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*sin(x) + b))/a

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maple [A]  time = 0.32, size = 21, normalized size = 1.11 \[ \frac {\ln \left (a +b \csc \relax (x )\right )}{a}-\frac {\ln \left (\csc \relax (x )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 11, normalized size = 0.58 \[ \frac {\log \left (a \sin \relax (x) + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*sin(x) + b)/a

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mupad [B]  time = 0.42, size = 55, normalized size = 2.89 \[ \frac {2\,\mathrm {atanh}\left (\frac {a\,\left (2\,b^3\,\sin \relax (x)+\frac {5\,a\,b^2}{2}-a^3-\frac {a\,b^2\,\cos \left (2\,x\right )}{2}\right )}{{\left (-a^2+\sin \relax (x)\,a\,b+2\,b^2\right )}^2}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + b/sin(x)),x)

[Out]

(2*atanh((a*(2*b^3*sin(x) + (5*a*b^2)/2 - a^3 - (a*b^2*cos(2*x))/2))/(2*b^2 - a^2 + a*b*sin(x))^2))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/(a + b*csc(x)), x)

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