3.432 \(\int \cos ^{\frac{3}{2}}(2 x) \sec ^3(x) \, dx\)

Optimal. Leaf size=49 \[ 2 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-\frac{5}{2} \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )-\frac{1}{2} \sqrt{\cos (2 x)} \tan (x) \sec (x) \]

[Out]

2*Sqrt[2]*ArcSin[Sqrt[2]*Sin[x]] - (5*ArcTan[Sin[x]/Sqrt[Cos[2*x]]])/2 - (Sqrt[Cos[2*x]]*Sec[x]*Tan[x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0574821, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {4364, 413, 523, 216, 377, 203} \[ 2 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-\frac{5}{2} \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )-\frac{1}{2} \sqrt{\cos (2 x)} \tan (x) \sec (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]^(3/2)*Sec[x]^3,x]

[Out]

2*Sqrt[2]*ArcSin[Sqrt[2]*Sin[x]] - (5*ArcTan[Sin[x]/Sqrt[Cos[2*x]]])/2 - (Sqrt[Cos[2*x]]*Sec[x]*Tan[x])/2

Rule 4364

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

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 216

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

Rule 377

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

Rule 203

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

Rubi steps

\begin{align*} \int \cos ^{\frac{3}{2}}(2 x) \sec ^3(x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-2 x^2\right )^{3/2}}{\left (1-x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=-\frac{1}{2} \sqrt{\cos (2 x)} \sec (x) \tan (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-3+8 x^2}{\sqrt{1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right )\\ &=-\frac{1}{2} \sqrt{\cos (2 x)} \sec (x) \tan (x)-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right )+4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-2 x^2}} \, dx,x,\sin (x)\right )\\ &=2 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-\frac{1}{2} \sqrt{\cos (2 x)} \sec (x) \tan (x)-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )\\ &=2 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-\frac{5}{2} \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )-\frac{1}{2} \sqrt{\cos (2 x)} \sec (x) \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.092552, size = 49, normalized size = 1. \[ \frac{1}{2} \left (4 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-5 \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )-\sqrt{\cos (2 x)} \tan (x) \sec (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]^(3/2)*Sec[x]^3,x]

[Out]

(4*Sqrt[2]*ArcSin[Sqrt[2]*Sin[x]] - 5*ArcTan[Sin[x]/Sqrt[Cos[2*x]]] - Sqrt[Cos[2*x]]*Sec[x]*Tan[x])/2

________________________________________________________________________________________

Maple [B]  time = 0.067, size = 100, normalized size = 2. \begin{align*} -{\frac{1}{4\, \left ( \cos \left ( x \right ) \right ) ^{2}\sin \left ( x \right ) }\sqrt{ \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 4\,\sqrt{2}\arcsin \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-3 \right ) \left ( \cos \left ( x \right ) \right ) ^{2}-5\,\arctan \left ( 1/2\,{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{2}-2}{\sqrt{-2\, \left ( \sin \left ( x \right ) \right ) ^{4}+ \left ( \sin \left ( x \right ) \right ) ^{2}}}} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\sqrt{-2\, \left ( \sin \left ( x \right ) \right ) ^{4}+ \left ( \sin \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)^(3/2)/cos(x)^3,x)

[Out]

-1/4*((2*cos(x)^2-1)*sin(x)^2)^(1/2)*(4*2^(1/2)*arcsin(4*cos(x)^2-3)*cos(x)^2-5*arctan(1/2*(3*cos(x)^2-2)/(-2*
sin(x)^4+sin(x)^2)^(1/2))*cos(x)^2+2*(-2*sin(x)^4+sin(x)^2)^(1/2))/cos(x)^2/sin(x)/(2*cos(x)^2-1)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (2 \, x\right )^{\frac{3}{2}}}{\cos \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^(3/2)/cos(x)^3,x, algorithm="maxima")

[Out]

integrate(cos(2*x)^(3/2)/cos(x)^3, x)

________________________________________________________________________________________

Fricas [B]  time = 3.84434, size = 365, normalized size = 7.45 \begin{align*} -\frac{2 \, \sqrt{2} \arctan \left (\frac{{\left (32 \, \sqrt{2} \cos \left (x\right )^{4} - 48 \, \sqrt{2} \cos \left (x\right )^{2} + 17 \, \sqrt{2}\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1}}{8 \,{\left (8 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} - 5 \, \arctan \left (\frac{3 \, \cos \left (x\right )^{2} - 2}{2 \, \sqrt{2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} + 2 \, \sqrt{2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^(3/2)/cos(x)^3,x, algorithm="fricas")

[Out]

-1/4*(2*sqrt(2)*arctan(1/8*(32*sqrt(2)*cos(x)^4 - 48*sqrt(2)*cos(x)^2 + 17*sqrt(2))*sqrt(2*cos(x)^2 - 1)/((8*c
os(x)^4 - 10*cos(x)^2 + 3)*sin(x)))*cos(x)^2 - 5*arctan(1/2*(3*cos(x)^2 - 2)/(sqrt(2*cos(x)^2 - 1)*sin(x)))*co
s(x)^2 + 2*sqrt(2*cos(x)^2 - 1)*sin(x))/cos(x)^2

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)**(3/2)/cos(x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (2 \, x\right )^{\frac{3}{2}}}{\cos \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^(3/2)/cos(x)^3,x, algorithm="giac")

[Out]

integrate(cos(2*x)^(3/2)/cos(x)^3, x)