3.1.40 \(\int (a \cos ^2(x))^{3/2} \, dx\) [40]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3282, 3286, 2717} \begin {gather*} \frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac {2}{3} a \tan (x) \sqrt {a \cos ^2(x)} \end {gather*}

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 2717

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

Rule 3282

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 3286

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
]])

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*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.76 \begin {gather*} \frac {1}{12} a \sqrt {a \cos ^2(x)} \sec (x) (9 \sin (x)+\sin (3 x)) \end {gather*}

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

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Maple [A]
time = 0.03, size = 24, normalized size = 0.71

method	result	size
default	\(\frac {a^{2} \cos \left (x \right ) \sin \left (x \right ) \left (\cos ^{2}\left (x \right )+2\right )}{3 \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\)	\(24\)
risch	\(-\frac {i a \,{\mathrm e}^{4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {3 i a \,{\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {3 i a \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {i a \,{\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}+24}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 17, normalized size = 0.50 \begin {gather*} \frac {1}{12} \, {\left (a \sin \left (3 \, x\right ) + 9 \, a \sin \left (x\right )\right )} \sqrt {a} \end {gather*}

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)

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Fricas [A]
time = 0.37, size = 26, normalized size = 0.76 \begin {gather*} \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 {gather*}

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.46, size = 17, normalized size = 0.50 \begin {gather*} -\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 {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (a\,{\cos \left (x\right )}^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cos(x)^2)^(3/2), x)

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