3.903 \(\int \frac{\sec (x) \tan ^2(x)}{4+3 \sec (x)} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan (x)}{3}-\frac{4}{9} \tanh ^{-1}(\sin (x))-\frac{1}{9} \sqrt{7} \log \left (\sqrt{7} \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\frac{1}{9} \sqrt{7} \log \left (\sin \left (\frac{x}{2}\right )+\sqrt{7} \cos \left (\frac{x}{2}\right )\right ) \]

[Out]

(-4*ArcTanh[Sin[x]])/9 - (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] - Sin[x/2]])/9 + (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] + Sin[x/
2]])/9 + Tan[x]/3

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Rubi [A]  time = 0.246008, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {4397, 2723, 3056, 3001, 3770, 2659, 206} \[ \frac{\tan (x)}{3}-\frac{4}{9} \tanh ^{-1}(\sin (x))-\frac{1}{9} \sqrt{7} \log \left (\sqrt{7} \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\frac{1}{9} \sqrt{7} \log \left (\sin \left (\frac{x}{2}\right )+\sqrt{7} \cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]*Tan[x]^2)/(4 + 3*Sec[x]),x]

[Out]

(-4*ArcTanh[Sin[x]])/9 - (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] - Sin[x/2]])/9 + (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] + Sin[x/
2]])/9 + Tan[x]/3

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 2723

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

Rule 3056

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c +
d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*
(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ
[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3770

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec (x) \tan ^2(x)}{4+3 \sec (x)} \, dx &=\int \frac{\tan ^2(x)}{3+4 \cos (x)} \, dx\\ &=\int \frac{\left (1-\cos ^2(x)\right ) \sec ^2(x)}{3+4 \cos (x)} \, dx\\ &=\frac{\tan (x)}{3}+\frac{1}{3} \int \frac{(-4-3 \cos (x)) \sec (x)}{3+4 \cos (x)} \, dx\\ &=\frac{\tan (x)}{3}-\frac{4}{9} \int \sec (x) \, dx+\frac{7}{9} \int \frac{1}{3+4 \cos (x)} \, dx\\ &=-\frac{4}{9} \tanh ^{-1}(\sin (x))+\frac{\tan (x)}{3}+\frac{14}{9} \operatorname{Subst}\left (\int \frac{1}{7-x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{4}{9} \tanh ^{-1}(\sin (x))-\frac{1}{9} \sqrt{7} \log \left (\sqrt{7} \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\frac{1}{9} \sqrt{7} \log \left (\sqrt{7} \cos \left (\frac{x}{2}\right )+\sin \left (\frac{x}{2}\right )\right )+\frac{\tan (x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0782563, size = 63, normalized size = 0.85 \[ \frac{1}{9} \left (3 \tan (x)+2 \sqrt{7} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{7}}\right )+4 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-4 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]*Tan[x]^2)/(4 + 3*Sec[x]),x]

[Out]

(2*Sqrt[7]*ArcTanh[Tan[x/2]/Sqrt[7]] + 4*Log[Cos[x/2] - Sin[x/2]] - 4*Log[Cos[x/2] + Sin[x/2]] + 3*Tan[x])/9

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Maple [A]  time = 0.021, size = 55, normalized size = 0.7 \begin{align*} -{\frac{1}{3} \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{4}{9}\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{4}{9}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{2\,\sqrt{7}}{9}{\it Artanh} \left ({\frac{\sqrt{7}}{7}\tan \left ({\frac{x}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x)^2/(4+3*sec(x)),x)

[Out]

-1/3/(1+tan(1/2*x))-4/9*ln(1+tan(1/2*x))-1/3/(tan(1/2*x)-1)+4/9*ln(tan(1/2*x)-1)+2/9*7^(1/2)*arctanh(1/7*tan(1
/2*x)*7^(1/2))

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Maxima [A]  time = 1.43588, size = 123, normalized size = 1.66 \begin{align*} -\frac{1}{9} \, \sqrt{7} \log \left (-\frac{\sqrt{7} - \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}{\sqrt{7} + \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right ) - \frac{2 \, \sin \left (x\right )}{3 \,{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}} - \frac{4}{9} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \frac{4}{9} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)^2/(4+3*sec(x)),x, algorithm="maxima")

[Out]

-1/9*sqrt(7)*log(-(sqrt(7) - sin(x)/(cos(x) + 1))/(sqrt(7) + sin(x)/(cos(x) + 1))) - 2/3*sin(x)/((sin(x)^2/(co
s(x) + 1)^2 - 1)*(cos(x) + 1)) - 4/9*log(sin(x)/(cos(x) + 1) + 1) + 4/9*log(sin(x)/(cos(x) + 1) - 1)

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Fricas [A]  time = 2.45333, size = 274, normalized size = 3.7 \begin{align*} \frac{\sqrt{7} \cos \left (x\right ) \log \left (\frac{2 \, \cos \left (x\right )^{2} + 2 \,{\left (3 \, \sqrt{7} \cos \left (x\right ) + 4 \, \sqrt{7}\right )} \sin \left (x\right ) + 24 \, \cos \left (x\right ) + 23}{16 \, \cos \left (x\right )^{2} + 24 \, \cos \left (x\right ) + 9}\right ) - 4 \, \cos \left (x\right ) \log \left (\sin \left (x\right ) + 1\right ) + 4 \, \cos \left (x\right ) \log \left (-\sin \left (x\right ) + 1\right ) + 6 \, \sin \left (x\right )}{18 \, \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)^2/(4+3*sec(x)),x, algorithm="fricas")

[Out]

1/18*(sqrt(7)*cos(x)*log((2*cos(x)^2 + 2*(3*sqrt(7)*cos(x) + 4*sqrt(7))*sin(x) + 24*cos(x) + 23)/(16*cos(x)^2
+ 24*cos(x) + 9)) - 4*cos(x)*log(sin(x) + 1) + 4*cos(x)*log(-sin(x) + 1) + 6*sin(x))/cos(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{2}{\left (x \right )} \sec{\left (x \right )}}{3 \sec{\left (x \right )} + 4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)**2/(4+3*sec(x)),x)

[Out]

Integral(tan(x)**2*sec(x)/(3*sec(x) + 4), x)

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Giac [A]  time = 1.22474, size = 97, normalized size = 1.31 \begin{align*} -\frac{1}{9} \, \sqrt{7} \log \left (\frac{{\left | -2 \, \sqrt{7} + 2 \, \tan \left (\frac{1}{2} \, x\right ) \right |}}{{\left | 2 \, \sqrt{7} + 2 \, \tan \left (\frac{1}{2} \, x\right ) \right |}}\right ) - \frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 1\right )}} - \frac{4}{9} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) + \frac{4}{9} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)^2/(4+3*sec(x)),x, algorithm="giac")

[Out]

-1/9*sqrt(7)*log(abs(-2*sqrt(7) + 2*tan(1/2*x))/abs(2*sqrt(7) + 2*tan(1/2*x))) - 2/3*tan(1/2*x)/(tan(1/2*x)^2
- 1) - 4/9*log(abs(tan(1/2*x) + 1)) + 4/9*log(abs(tan(1/2*x) - 1))