3.1.19 \(\int (c \cos (a+b x))^{3/2} \, dx\) [19]

Optimal. Leaf size=70 \[ \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b} \]

[Out]

2/3*c^2*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)
/b/(c*cos(b*x+a))^(1/2)+2/3*c*sin(b*x+a)*(c*cos(b*x+a))^(1/2)/b

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \begin {gather*} \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sin (a+b x) \sqrt {c \cos (a+b x)}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(3/2),x]

[Out]

(2*c^2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(3*b*Sqrt[c*Cos[a + b*x]]) + (2*c*Sqrt[c*Cos[a + b*x]]*Si
n[a + b*x])/(3*b)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps

\begin {align*} \int (c \cos (a+b x))^{3/2} \, dx &=\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} c^2 \int \frac {1}{\sqrt {c \cos (a+b x)}} \, dx\\ &=\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac {\left (c^2 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{3 \sqrt {c \cos (a+b x)}}\\ &=\frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 58, normalized size = 0.83 \begin {gather*} \frac {2 (c \cos (a+b x))^{3/2} \left (F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (a+b x)\right )}{3 b \cos ^{\frac {3}{2}}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(3/2),x]

[Out]

(2*(c*Cos[a + b*x])^(3/2)*(EllipticF[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*Sin[a + b*x]))/(3*b*Cos[a + b*x]^(3/
2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(86)=172\).
time = 0.04, size = 190, normalized size = 2.71

method	result	size
default	\(-\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c^{2} \left (4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(c*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*c^2*(4*sin(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2*a)-
2*sin(1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a)+(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*Ellipt
icF(cos(1/2*b*x+1/2*a),2^(1/2)))/(-c*(2*sin(1/2*b*x+1/2*a)^4-sin(1/2*b*x+1/2*a)^2))^(1/2)/sin(1/2*b*x+1/2*a)/(
c*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(3/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 77, normalized size = 1.10 \begin {gather*} \frac {-i \, \sqrt {2} c^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} c^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} c \sin \left (b x + a\right )}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-I*sqrt(2)*c^(3/2)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) + I*sqrt(2)*c^(3/2)*weierstr
assPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) + 2*sqrt(c*cos(b*x + a))*c*sin(b*x + a))/b

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(b*x+a))**(3/2),x)

[Out]

Integral((c*cos(a + b*x))**(3/2), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(3/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,\cos \left (a+b\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(a + b*x))^(3/2),x)

[Out]

int((c*cos(a + b*x))^(3/2), x)

________________________________________________________________________________________