3.386 \(\int \sec (3 x) \sin ^3(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{3} \log (\cos (x))-\frac{1}{24} \log \left (3-4 \cos ^2(x)\right ) \]

[Out]

Log[Cos[x]]/3 - Log[3 - 4*Cos[x]^2]/24

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Rubi [A]  time = 0.041225, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4366, 446, 72} \[ \frac{1}{3} \log (\cos (x))-\frac{1}{24} \log \left (3-4 \cos ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sec[3*x]*Sin[x]^3,x]

[Out]

Log[Cos[x]]/3 - Log[3 - 4*Cos[x]^2]/24

Rule 4366

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dis
t[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \sec (3 x) \sin ^3(x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+x^2}{x \left (3-4 x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x}{(3-4 x) x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{3 x}+\frac{1}{3 (-3+4 x)}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac{1}{3} \log (\cos (x))-\frac{1}{24} \log \left (3-4 \cos ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0153123, size = 21, normalized size = 1. \[ \frac{1}{3} \log (\cos (x))-\frac{1}{24} \log \left (1-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[3*x]*Sin[x]^3,x]

[Out]

Log[Cos[x]]/3 - Log[1 - 4*Sin[x]^2]/24

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Maple [A]  time = 0.053, size = 18, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \cos \left ( x \right ) \right ) }{3}}-{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-3 \right ) }{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/cos(3*x),x)

[Out]

1/3*ln(cos(x))-1/24*ln(4*cos(x)^2-3)

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Maxima [B]  time = 1.4324, size = 109, normalized size = 5.19 \begin{align*} -\frac{1}{48} \, \log \left (-2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac{1}{6} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/cos(3*x),x, algorithm="maxima")

[Out]

-1/48*log(-2*(cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 + sin(4*x)^2 - 2*sin(4*x)*sin(2*x) + sin(2*x)^2
- 2*cos(2*x) + 1) + 1/6*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)

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Fricas [A]  time = 2.32023, size = 62, normalized size = 2.95 \begin{align*} -\frac{1}{24} \, \log \left (4 \, \cos \left (x\right )^{2} - 3\right ) + \frac{1}{3} \, \log \left (-\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/cos(3*x),x, algorithm="fricas")

[Out]

-1/24*log(4*cos(x)^2 - 3) + 1/3*log(-cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/cos(3*x),x)

[Out]

Integral(sin(x)**3/cos(3*x), x)

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Giac [B]  time = 1.1141, size = 120, normalized size = 5.71 \begin{align*} -\frac{1}{8} \, \log \left (-\frac{\cos \left (x\right ) + 1}{\cos \left (x\right ) - 1} - \frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 2\right ) + \frac{1}{6} \, \log \left (-\frac{\cos \left (x\right ) + 1}{\cos \left (x\right ) - 1} - \frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 2\right ) - \frac{1}{24} \, \log \left ({\left | -\frac{\cos \left (x\right ) + 1}{\cos \left (x\right ) - 1} - \frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 14 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/cos(3*x),x, algorithm="giac")

[Out]

-1/8*log(-(cos(x) + 1)/(cos(x) - 1) - (cos(x) - 1)/(cos(x) + 1) + 2) + 1/6*log(-(cos(x) + 1)/(cos(x) - 1) - (c
os(x) - 1)/(cos(x) + 1) - 2) - 1/24*log(abs(-(cos(x) + 1)/(cos(x) - 1) - (cos(x) - 1)/(cos(x) + 1) - 14))