3.421 \(\int \cos (x) (-\cos ^2(x)-5 \sin ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac{3}{8} \sin (x) \sqrt{-4 \sin ^2(x)-1}+\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{-4 \sin ^2(x)-1}}\right ) \]

[Out]

(3*ArcTan[(2*Sin[x])/Sqrt[-1 - 4*Sin[x]^2]])/16 - (3*Sin[x]*Sqrt[-1 - 4*Sin[x]^2])/8 + (Sin[x]*(-1 - 4*Sin[x]^
2)^(3/2))/4

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Rubi [A]  time = 0.051131, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4356, 195, 217, 203} \[ \frac{1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac{3}{8} \sin (x) \sqrt{-4 \sin ^2(x)-1}+\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{-4 \sin ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*(-Cos[x]^2 - 5*Sin[x]^2)^(3/2),x]

[Out]

(3*ArcTan[(2*Sin[x])/Sqrt[-1 - 4*Sin[x]^2]])/16 - (3*Sin[x]*Sqrt[-1 - 4*Sin[x]^2])/8 + (Sin[x]*(-1 - 4*Sin[x]^
2)^(3/2))/4

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, 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] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

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

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 (x) \left (-\cos ^2(x)-5 \sin ^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \left (-1-4 x^2\right )^{3/2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}-\frac{3}{4} \operatorname{Subst}\left (\int \sqrt{-1-4 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{3}{8} \sin (x) \sqrt{-1-4 \sin ^2(x)}+\frac{1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-4 x^2}} \, dx,x,\sin (x)\right )\\ &=-\frac{3}{8} \sin (x) \sqrt{-1-4 \sin ^2(x)}+\frac{1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\frac{\sin (x)}{\sqrt{-1-4 \sin ^2(x)}}\right )\\ &=\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{-1-4 \sin ^2(x)}}\right )-\frac{3}{8} \sin (x) \sqrt{-1-4 \sin ^2(x)}+\frac{1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0750157, size = 61, normalized size = 1.05 \[ \frac{\sqrt{2 \cos (2 x)-3} \left (2 (2 \sin (3 x)-11 \sin (x)) \sqrt{3-2 \cos (2 x)}-3 \sinh ^{-1}(2 \sin (x))\right )}{16 \sqrt{4 \sin ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*(-Cos[x]^2 - 5*Sin[x]^2)^(3/2),x]

[Out]

(Sqrt[-3 + 2*Cos[2*x]]*(-3*ArcSinh[2*Sin[x]] + 2*Sqrt[3 - 2*Cos[2*x]]*(-11*Sin[x] + 2*Sin[3*x])))/(16*Sqrt[1 +
4*Sin[x]^2])

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Maple [A]  time = 0.082, size = 82, normalized size = 1.4 \begin{align*}{\frac{1}{32\,\sin \left ( x \right ) }\sqrt{ \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-5 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( -32\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}- \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}-20\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}- \left ( \sin \left ( x \right ) \right ) ^{2}}+3\,\arcsin \left ( 8\, \left ( \sin \left ( x \right ) \right ) ^{2}+1 \right ) \right ){\frac{1}{\sqrt{4\, \left ( \cos \left ( x \right ) \right ) ^{2}-5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(-cos(x)^2-5*sin(x)^2)^(3/2),x)

[Out]

1/32*((4*cos(x)^2-5)*sin(x)^2)^(1/2)*(-32*(-4*sin(x)^4-sin(x)^2)^(1/2)*sin(x)^2-20*(-4*sin(x)^4-sin(x)^2)^(1/2
)+3*arcsin(8*sin(x)^2+1))/sin(x)/(4*cos(x)^2-5)^(1/2)

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Maxima [C]  time = 1.48356, size = 49, normalized size = 0.84 \begin{align*} \frac{1}{4} \,{\left (-4 \, \sin \left (x\right )^{2} - 1\right )}^{\frac{3}{2}} \sin \left (x\right ) - \frac{3}{8} \, \sqrt{-4 \, \sin \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac{3}{16} i \, \operatorname{arsinh}\left (2 \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2-5*sin(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/4*(-4*sin(x)^2 - 1)^(3/2)*sin(x) - 3/8*sqrt(-4*sin(x)^2 - 1)*sin(x) - 3/16*I*arcsinh(2*sin(x))

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Fricas [C]  time = 2.78899, size = 431, normalized size = 7.43 \begin{align*} \frac{1}{128} \,{\left (12 i \, e^{\left (4 i \, x\right )} \log \left (-\frac{1}{2} \, \sqrt{e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1}{\left (4 \, e^{\left (2 i \, x\right )} - 5\right )} + 2 \, e^{\left (4 i \, x\right )} - \frac{11}{2} \, e^{\left (2 i \, x\right )} + \frac{5}{2}\right ) - 12 i \, e^{\left (4 i \, x\right )} \log \left (\sqrt{e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} - 1\right ) +{\left (-16 i \, e^{\left (6 i \, x\right )} + 88 i \, e^{\left (4 i \, x\right )} - 88 i \, e^{\left (2 i \, x\right )} + 16 i\right )} \sqrt{e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1} - 145 i \, e^{\left (4 i \, x\right )}\right )} e^{\left (-4 i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2-5*sin(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/128*(12*I*e^(4*I*x)*log(-1/2*sqrt(e^(4*I*x) - 3*e^(2*I*x) + 1)*(4*e^(2*I*x) - 5) + 2*e^(4*I*x) - 11/2*e^(2*I
*x) + 5/2) - 12*I*e^(4*I*x)*log(sqrt(e^(4*I*x) - 3*e^(2*I*x) + 1) - e^(2*I*x) - 1) + (-16*I*e^(6*I*x) + 88*I*e
^(4*I*x) - 88*I*e^(2*I*x) + 16*I)*sqrt(e^(4*I*x) - 3*e^(2*I*x) + 1) - 145*I*e^(4*I*x))*e^(-4*I*x)

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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(x)*(-cos(x)**2-5*sin(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [C]  time = 1.11235, size = 41, normalized size = 0.71 \begin{align*} -\frac{1}{8} i \,{\left (8 \, \sin \left (x\right )^{2} + 5\right )} \sqrt{4 \, \sin \left (x\right )^{2} + 1} \sin \left (x\right ) - \frac{3}{16} i \, \arcsin \left (2 i \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2-5*sin(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/8*I*(8*sin(x)^2 + 5)*sqrt(4*sin(x)^2 + 1)*sin(x) - 3/16*I*arcsin(2*I*sin(x))