3.263 \(\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)} \, dx\)

Optimal. Leaf size=129 \[ -\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{4 d} \]

[Out]

(-3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*d) + (3*a*Sqrt[C
os[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) - (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a -
a*Cos[c + d*x]])

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Rubi [A]  time = 0.191758, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2770, 2775, 207} \[ -\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]],x]

[Out]

(-3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*d) + (3*a*Sqrt[C
os[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) - (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a -
a*Cos[c + d*x]])

Rule 2770

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d)
)/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2775

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*b)/f, Subst[Int[1/(b + d*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 207

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

Rubi steps

\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)} \, dx &=-\frac{a \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}-\frac{3}{4} \int \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \, dx\\ &=\frac{3 a \sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{a \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3}{8} \int \frac{\sqrt{a-a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{3 a \sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{a \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{4 d}\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{4 d}+\frac{3 a \sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{a \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 4.04623, size = 289, normalized size = 2.24 \[ -\frac{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \left (2 \sqrt{2} \left (\cos \left (\frac{3}{2} (c+d x)\right )-2 \cos \left (\frac{1}{2} (c+d x)\right )\right ) \csc \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+3 \sqrt{\cos (c)-i \sin (c)} \left (\cot \left (\frac{1}{2} (c+d x)\right )+i\right ) \tanh ^{-1}\left (\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right )+3 \sqrt{\cos (c)-i \sin (c)} \left (\cot \left (\frac{1}{2} (c+d x)\right )+i\right ) \tanh ^{-1}\left (\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right )\right )}{8 d \sqrt{i \sin (c) \left (-1+e^{2 i d x}\right )+\cos (c) \left (1+e^{2 i d x}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]],x]

[Out]

-(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]]*(3*ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((
2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + 3*ArcTanh[Sqrt[Co
s[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[
c] - I*Sin[c]] + 2*Sqrt[2]*(-2*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])*Csc[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Co
s[d*x] + I*Sin[d*x])]))/(8*d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]])

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Maple [A]  time = 0.404, size = 165, normalized size = 1.3 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( 2\,\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) -3\,\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+3\,{\it Artanh} \left ( \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \right ) \sqrt{-2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(3/2)*(a-cos(d*x+c)*a)^(1/2),x)

[Out]

1/8/d*2^(1/2)*(-1+cos(d*x+c))*(2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2-(cos(d*x+c)/(1+cos(d*x+c)))^(1
/2)*cos(d*x+c)-3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3*arctanh((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)))*(-2*a*(-1+cos
(d*x+c)))^(1/2)*cos(d*x+c)^(3/2)/(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)/sin(d*x+c)^3

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Maxima [B]  time = 2.06477, size = 1435, normalized size = 11.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(3/2)*(a-a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/16*(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*((cos(1/2*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c)))*sin(2*d*x + 2*c) - (cos(2*d*x + 2*c) - 2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*
x + 2*c))) - sin(2*d*x + 2*c))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + ((cos(2*d*x + 2*c) -
2)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(2*d*x + 2*c)*sin(1/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + cos(2*d*x + 2*c) - 2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(-a)
+ 3*sqrt(-a)*(arctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(1/2*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - cos(1/2*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))), (cos(2*d*x
+ 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c) + 1))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c) + 1))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))) + 1) - arctan2((cos(2*d*x + 2*c)^2 + sin(2*
d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))*sin(1/2*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(1
/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c
) + 1)^(1/4)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2
*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c)))) - 1) + arctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1
/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 1) - arctan2((cos(2*d*x + 2*c)^2 +
sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (
cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c) + 1)) - 1)))/d

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Fricas [A]  time = 2.21998, size = 417, normalized size = 3.23 \begin{align*} \frac{3 \, \sqrt{a} \log \left (\frac{4 \, \sqrt{-a \cos \left (d x + c\right ) + a}{\left (2 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \sqrt{\cos \left (d x + c\right )} -{\left (8 \, a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 4 \, \sqrt{-a \cos \left (d x + c\right ) + a}{\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 3\right )} \sqrt{\cos \left (d x + c\right )}}{16 \, d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(3/2)*(a-a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/16*(3*sqrt(a)*log((4*sqrt(-a*cos(d*x + c) + a)*(2*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*sqrt(cos(d*x
+ c)) - (8*a*cos(d*x + c)^2 + 8*a*cos(d*x + c) + a)*sin(d*x + c))/sin(d*x + c))*sin(d*x + c) - 4*sqrt(-a*cos(d
*x + c) + a)*(2*cos(d*x + c)^2 - cos(d*x + c) - 3)*sqrt(cos(d*x + c)))/(d*sin(d*x + c))

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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(d*x+c)**(3/2)*(a-a*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [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(d*x+c)^(3/2)*(a-a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out