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

Optimal. Leaf size=66 \[ \frac{b^2 \log (a+b \csc (x))}{a \left (a^2-b^2\right )}-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (\csc (x)+1)}{2 (a-b)}-\frac{\log (\sin (x))}{a} \]

[Out]

-Log[1 - Csc[x]]/(2*(a + b)) - Log[1 + Csc[x]]/(2*(a - b)) + (b^2*Log[a + b*Csc[x]])/(a*(a^2 - b^2)) - Log[Sin
[x]]/a

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Rubi [A]  time = 0.0906753, antiderivative size = 66, 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 = {3885, 894} \[ \frac{b^2 \log (a+b \csc (x))}{a \left (a^2-b^2\right )}-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (\csc (x)+1)}{2 (a-b)}-\frac{\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/(a + b*Csc[x]),x]

[Out]

-Log[1 - Csc[x]]/(2*(a + b)) - Log[1 + Csc[x]]/(2*(a - b)) + (b^2*Log[a + b*Csc[x]])/(a*(a^2 - b^2)) - Log[Sin
[x]]/a

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]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.0611355, size = 56, normalized size = 0.85 \[ -\frac{-2 b^2 \log (a \sin (x)+b)+a (a-b) \log (1-\sin (x))+a (a+b) \log (\sin (x)+1)}{2 a (a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/(a + b*Csc[x]),x]

[Out]

-(a*(a - b)*Log[1 - Sin[x]] + a*(a + b)*Log[1 + Sin[x]] - 2*b^2*Log[b + a*Sin[x]])/(2*a*(a - b)*(a + b))

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Maple [A]  time = 0.049, size = 60, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}\ln \left ( b+a\sin \left ( x \right ) \right ) }{ \left ( a+b \right ) \left ( a-b \right ) a}}-{\frac{\ln \left ( \sin \left ( x \right ) +1 \right ) }{2\,a-2\,b}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{2\,a+2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2/(a+b)/(a-b)/a*ln(b+a*sin(x))-1/(2*a-2*b)*ln(sin(x)+1)-1/(2*a+2*b)*ln(sin(x)-1)

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Maxima [A]  time = 0.976723, size = 68, normalized size = 1.03 \begin{align*} \frac{b^{2} \log \left (a \sin \left (x\right ) + b\right )}{a^{3} - a b^{2}} - \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.551987, size = 143, normalized size = 2.17 \begin{align*} \frac{2 \, b^{2} \log \left (a \sin \left (x\right ) + b\right ) -{\left (a^{2} + a b\right )} \log \left (\sin \left (x\right ) + 1\right ) -{\left (a^{2} - a b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a^{3} - a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(x)/(a + b*csc(x)), x)

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Giac [A]  time = 1.40585, size = 72, normalized size = 1.09 \begin{align*} \frac{b^{2} \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{3} - a b^{2}} - \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(abs(a*sin(x) + b))/(a^3 - a*b^2) - 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(-sin(x) + 1)/(a + b)