∫ tan(x)_____∘ 1 + sec3(x) dx [7]

Optimal. Leaf size=15 \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+\sec ^3(x)}\right ) \]

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {4224, 272, 65, 213} \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {\sec ^3(x)+1}\right ) \end {gather*}

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,

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

, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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]]

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With

[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x)

, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[

n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+\sec ^3(x)}\right ) \end {gather*}

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method	result	size

derivativedivides	\(-\frac {2 \arctanh \left (\sqrt {1+\sec ^{3}\left (x \right )}\right )}{3}\)	\(12\)
default	\(-\frac {2 \arctanh \left (\sqrt {1+\sec ^{3}\left (x \right )}\right )}{3}\)	\(12\)

int(tan(x)/(1+sec(x)^3)^(1/2),x,method=_RETURNVERBOSE)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (11) = 22\).
time = 2.68, size = 27, normalized size = 1.80 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {\frac {1}{\cos \left (x\right )^{3}} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {\frac {1}{\cos \left (x\right )^{3}} + 1} - 1\right ) \end {gather*}

integrate(tan(x)/(1+sec(x)^3)^(1/2),x, algorithm="maxima")

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (11) = 22\).
time = 0.62, size = 30, normalized size = 2.00 \begin {gather*} \frac {1}{3} \, \log \left (2 \, \sqrt {\frac {\cos \left (x\right )^{3} + 1}{\cos \left (x\right )^{3}}} \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{3} - 1\right ) \end {gather*}

integrate(tan(x)/(1+sec(x)^3)^(1/2),x, algorithm="fricas")

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (x \right )}}{\sqrt {\left (\sec {\left (x \right )} + 1\right ) \left (\sec ^{2}{\left (x \right )} - \sec {\left (x \right )} + 1\right )}}\, dx \end {gather*}

Integral(tan(x)/sqrt((sec(x) + 1)*(sec(x)**2 - sec(x) + 1)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
time = 0.47, size = 28, normalized size = 1.87 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {\frac {1}{\cos \left (x\right )^{3}} + 1} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \sqrt {\frac {1}{\cos \left (x\right )^{3}} + 1} - 1 \right |}\right ) \end {gather*}

integrate(tan(x)/(1+sec(x)^3)^(1/2),x, algorithm="giac")

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {\mathrm {tan}\left (x\right )}{\sqrt {\frac {1}{{\cos \left (x\right )}^3}+1}} \,d x \end {gather*}

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3.1.7 \(\int \frac {\tan (x)}{\sqrt {1+\sec ^3(x)}} \, dx\) [7]