3.11 \(\int \frac{\tan ^3(x)}{a+b \cos (x)} \, dx\)

Optimal. Leaf size=57 \[ \frac{\left (a^2-b^2\right ) \log (\cos (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{\sec ^2(x)}{2 a} \]

[Out]

((a^2 - b^2)*Log[Cos[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cos[x]])/a^3 - (b*Sec[x])/a^2 + Sec[x]^2/(2*a)

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Rubi [A]  time = 0.0835597, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2721, 894} \[ \frac{\left (a^2-b^2\right ) \log (\cos (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{\sec ^2(x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^3/(a + b*Cos[x]),x]

[Out]

((a^2 - b^2)*Log[Cos[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cos[x]])/a^3 - (b*Sec[x])/a^2 + Sec[x]^2/(2*a)

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

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 ^3(x)}{a+b \cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{b^2-x^2}{x^3 (a+x)} \, dx,x,b \cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{b^2}{a x^3}-\frac{b^2}{a^2 x^2}+\frac{-a^2+b^2}{a^3 x}+\frac{a^2-b^2}{a^3 (a+x)}\right ) \, dx,x,b \cos (x)\right )\\ &=\frac{\left (a^2-b^2\right ) \log (\cos (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{\sec ^2(x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.11586, size = 46, normalized size = 0.81 \[ \frac{2 \left (a^2-b^2\right ) (\log (\cos (x))-\log (a+b \cos (x)))+a^2 \sec ^2(x)-2 a b \sec (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^3/(a + b*Cos[x]),x]

[Out]

(2*(a^2 - b^2)*(Log[Cos[x]] - Log[a + b*Cos[x]]) - 2*a*b*Sec[x] + a^2*Sec[x]^2)/(2*a^3)

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Maple [A]  time = 0.069, size = 65, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ) }{a}}+{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ){b}^{2}}{{a}^{3}}}-{\frac{b}{{a}^{2}\cos \left ( x \right ) }}+{\frac{\ln \left ( \cos \left ( x \right ) \right ) }{a}}-{\frac{\ln \left ( \cos \left ( x \right ) \right ){b}^{2}}{{a}^{3}}}+{\frac{1}{2\,a \left ( \cos \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3/(a+b*cos(x)),x)

[Out]

-ln(a+b*cos(x))/a+1/a^3*ln(a+b*cos(x))*b^2-1/a^2*b/cos(x)+ln(cos(x))/a-1/a^3*ln(cos(x))*b^2+1/2/a/cos(x)^2

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Maxima [A]  time = 0.991367, size = 76, normalized size = 1.33 \begin{align*} -\frac{{\left (a^{2} - b^{2}\right )} \log \left (b \cos \left (x\right ) + a\right )}{a^{3}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (\cos \left (x\right )\right )}{a^{3}} - \frac{2 \, b \cos \left (x\right ) - a}{2 \, a^{2} \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46932, size = 167, normalized size = 2.93 \begin{align*} -\frac{2 \,{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} \log \left (-b \cos \left (x\right ) - a\right ) - 2 \,{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right )\right ) + 2 \, a b \cos \left (x\right ) - a^{2}}{2 \, a^{3} \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (x \right )}}{a + b \cos{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**3/(a+b*cos(x)),x)

[Out]

Integral(tan(x)**3/(a + b*cos(x)), x)

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Giac [A]  time = 1.33934, size = 89, normalized size = 1.56 \begin{align*} \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | \cos \left (x\right ) \right |}\right )}{a^{3}} - \frac{{\left (a^{2} b - b^{3}\right )} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a^{3} b} - \frac{2 \, a b \cos \left (x\right ) - a^{2}}{2 \, a^{3} \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="giac")

[Out]

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