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

Optimal. Leaf size=28 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3230, 266, 63, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/Sqrt[a + b*Cos[x]^3],x]

[Out]

(2*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]])/(3*Sqrt[a])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rubi steps

\begin {align*} \int \frac {\tan (x)}{\sqrt {a+b \cos ^3(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^3}} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cos ^3(x)\right )\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cos ^3(x)}\right )}{3 b}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.00 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/Sqrt[a + b*Cos[x]^3],x]

[Out]

(2*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]])/(3*Sqrt[a])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)^3)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   failed of mode Union(SparseUnivariatePol
ynomial(Expression(Complex(Integer))),failed) cannot be coerced to mode SparseUnivariatePolynomial(Expression(
Complex(Integer)))

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giac [A]  time = 0.41, size = 24, normalized size = 0.86 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {b \cos \relax (x)^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)^3)^(1/2),x, algorithm="giac")

[Out]

-2/3*arctan(sqrt(b*cos(x)^3 + a)/sqrt(-a))/sqrt(-a)

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maple [A]  time = 0.12, size = 21, normalized size = 0.75 \[ \frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cos ^{3}\relax (x )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(a+b*cos(x)^3)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.76, size = 39, normalized size = 1.39 \[ -\frac {\log \left (\frac {\sqrt {b \cos \relax (x)^{3} + a} - \sqrt {a}}{\sqrt {b \cos \relax (x)^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)^3)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {tan}\relax (x)}{\sqrt {b\,{\cos \relax (x)}^3+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(a + b*cos(x)^3)^(1/2),x)

[Out]

int(tan(x)/(a + b*cos(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\relax (x )}}{\sqrt {a + b \cos ^{3}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)**3)**(1/2),x)

[Out]

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

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