3.3 \(\int \frac{\tan ^2(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=15 \[ \frac{\tan (x)}{a}-\frac{\tanh ^{-1}(\sin (x))}{a} \]

[Out]

-(ArcTanh[Sin[x]]/a) + Tan[x]/a

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Rubi [A]  time = 0.0501199, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 3767, 8, 3770} \[ \frac{\tan (x)}{a}-\frac{\tanh ^{-1}(\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^2/(a + a*Cos[x]),x]

[Out]

-(ArcTanh[Sin[x]]/a) + Tan[x]/a

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tan ^2(x)}{a+a \cos (x)} \, dx &=-\frac{\int \sec (x) \, dx}{a}+\frac{\int \sec ^2(x) \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\sin (x))}{a}-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (x))}{a}\\ &=-\frac{\tanh ^{-1}(\sin (x))}{a}+\frac{\tan (x)}{a}\\ \end{align*}

Mathematica [B]  time = 0.0707014, size = 39, normalized size = 2.6 \[ \frac{\tan (x)+\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^2/(a + a*Cos[x]),x]

[Out]

(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Tan[x])/a

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Maple [B]  time = 0.053, size = 51, normalized size = 3.4 \begin{align*} -{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^2/(a+a*cos(x)),x)

[Out]

-1/a/(tan(1/2*x)-1)+1/a*ln(tan(1/2*x)-1)-1/a/(tan(1/2*x)+1)-1/a*ln(tan(1/2*x)+1)

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Maxima [B]  time = 1.14815, size = 82, normalized size = 5.47 \begin{align*} -\frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}{a} + \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right )}{a} + \frac{2 \, \sin \left (x\right )}{{\left (a - \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^2/(a+a*cos(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.36594, size = 107, normalized size = 7.13 \begin{align*} -\frac{\cos \left (x\right ) \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right ) \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, \sin \left (x\right )}{2 \, a \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^2/(a+a*cos(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{2}{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**2/(a+a*cos(x)),x)

[Out]

Integral(tan(x)**2/(cos(x) + 1), x)/a

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Giac [B]  time = 1.27762, size = 61, normalized size = 4.07 \begin{align*} -\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right )}{a} + \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right )}{a} - \frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 1\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^2/(a+a*cos(x)),x, algorithm="giac")

[Out]

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