3.40 \(\int (a \cos ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac{2}{3} a \tan (x) \sqrt{a \cos ^2(x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0268519, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3203, 3207, 2637} \[ \frac{1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac{2}{3} a \tan (x) \sqrt{a \cos ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (a \cos ^2(x)\right )^{3/2} \, dx &=\frac{1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac{1}{3} (2 a) \int \sqrt{a \cos ^2(x)} \, dx\\ &=\frac{1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac{1}{3} \left (2 a \sqrt{a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\frac{2}{3} a \sqrt{a \cos ^2(x)} \tan (x)+\frac{1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0094787, size = 26, normalized size = 0.76 \[ \frac{1}{12} a (9 \sin (x)+\sin (3 x)) \sec (x) \sqrt{a \cos ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a*Cos[x]^2]*Sec[x]*(9*Sin[x] + Sin[3*x]))/12

________________________________________________________________________________________

Maple [A]  time = 0.605, size = 24, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}\cos \left ( x \right ) \sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}+2 \right ) }{3}{\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.55931, size = 23, normalized size = 0.68 \begin{align*} \frac{1}{12} \,{\left (a \sin \left (3 \, x\right ) + 9 \, a \sin \left (x\right )\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a*sin(3*x) + 9*a*sin(x))*sqrt(a)

________________________________________________________________________________________

Fricas [A]  time = 1.04613, size = 74, normalized size = 2.18 \begin{align*} \frac{{\left (a \cos \left (x\right )^{2} + 2 \, a\right )} \sqrt{a \cos \left (x\right )^{2}} \sin \left (x\right )}{3 \, \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*cos(x)^2 + 2*a)*sqrt(a*cos(x)^2)*sin(x)/cos(x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.4166, size = 23, normalized size = 0.68 \begin{align*} -\frac{1}{3} \,{\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} a^{\frac{3}{2}} \mathrm{sgn}\left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sin(x)^3 - 3*sin(x))*a^(3/2)*sgn(cos(x))