3.4 \(\int \frac{\tan (x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{1}{2 a (\sin (x)+1)}-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (\sin (x)+1)}{4 a} \]

[Out]

-Log[1 - Sin[x]]/(4*a) - (3*Log[1 + Sin[x]])/(4*a) - 1/(2*a*(1 + Sin[x]))

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Rubi [A]  time = 0.0486917, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 88} \[ -\frac{1}{2 a (\sin (x)+1)}-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (\sin (x)+1)}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/(a + a*Csc[x]),x]

[Out]

-Log[1 - Sin[x]]/(4*a) - (3*Log[1 + Sin[x]])/(4*a) - 1/(2*a*(1 + Sin[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]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\tan (x)}{a+a \csc (x)} \, dx &=a^2 \operatorname{Subst}\left (\int \frac{x^2}{(a-a x) (a+a x)^2} \, dx,x,\sin (x)\right )\\ &=a^2 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^3 (-1+x)}+\frac{1}{2 a^3 (1+x)^2}-\frac{3}{4 a^3 (1+x)}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (1+\sin (x))}{4 a}-\frac{1}{2 a (1+\sin (x))}\\ \end{align*}

Mathematica [A]  time = 0.0539248, size = 30, normalized size = 0.75 \[ -\frac{\frac{2}{\sin (x)+1}+\log (1-\sin (x))+3 \log (\sin (x)+1)}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/(a + a*Csc[x]),x]

[Out]

-(Log[1 - Sin[x]] + 3*Log[1 + Sin[x]] + 2/(1 + Sin[x]))/(4*a)

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Maple [A]  time = 0.062, size = 33, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,a \left ( \sin \left ( x \right ) +1 \right ) }}-{\frac{3\,\ln \left ( \sin \left ( x \right ) +1 \right ) }{4\,a}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a/(sin(x)+1)-3/4*ln(sin(x)+1)/a-1/4/a*ln(sin(x)-1)

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Maxima [A]  time = 0.957275, size = 42, normalized size = 1.05 \begin{align*} -\frac{3 \, \log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{4 \, a} - \frac{1}{2 \,{\left (a \sin \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*log(sin(x) + 1)/a - 1/4*log(sin(x) - 1)/a - 1/2/(a*sin(x) + a)

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Fricas [A]  time = 0.50747, size = 122, normalized size = 3.05 \begin{align*} -\frac{3 \,{\left (\sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right ) + 1\right ) +{\left (\sin \left (x\right ) + 1\right )} \log \left (-\sin \left (x\right ) + 1\right ) + 2}{4 \,{\left (a \sin \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*(sin(x) + 1)*log(sin(x) + 1) + (sin(x) + 1)*log(-sin(x) + 1) + 2)/(a*sin(x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(x)/(csc(x) + 1), x)/a

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Giac [A]  time = 1.35769, size = 46, normalized size = 1.15 \begin{align*} -\frac{3 \, \log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{1}{2 \, a{\left (\sin \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*log(sin(x) + 1)/a - 1/4*log(-sin(x) + 1)/a - 1/2/(a*(sin(x) + 1))