3.447 \(\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx\)
Optimal. Leaf size=98 \[ -\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{3 d} \]
[Out]
(-2*Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqr
t[-Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)
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Rubi [A] time = 0.201563, antiderivative size = 98, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used =
{2995, 2994} \[ -\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]),x]
[Out]
(-2*Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqr
t[-Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)
Rule 2995
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> -Dist[Sqrt[-(b*Sin[e + f*x])]/Sqrt[b*Sin[e + f*x]], Int[(A + B*Sin[e + f*x])/((-
(b*Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2,
0] && EqQ[A, B] && NegQ[(c + d)/b]
Rule 2994
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c
- d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[
(c + d)/b, 2])], -((c + d)/(c - d))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] &&
EqQ[A, B] && PosQ[(c + d)/b]
Rubi steps
\begin{align*} \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx &=-\frac{\sqrt{-\cos (c+d x)} \int \frac{1+\cos (c+d x)}{(-\cos (c+d x))^{3/2} \sqrt{-3+2 \cos (c+d x)}} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{-3+2 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right ) \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)}}{3 d}\\ \end{align*}
Mathematica [F] time = 40.906, size = 0, normalized size = 0. \[ \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]),x]
[Out]
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]), x]
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Maple [B] time = 0.352, size = 714, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x)
[Out]
1/15/d/(-3+2*cos(d*x+c))^(1/2)*(3*I*(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3
/2)*EllipticF(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))*5^(1/2)*2^(1/2)*sin(d*x+c)*cos(d*x+c)^2+6*I*
(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*EllipticF(I*(-1+cos(d*x+c))*5^(1
/2)/sin(d*x+c),1/5*I*5^(1/2))*5^(1/2)*2^(1/2)*sin(d*x+c)*cos(d*x+c)+5*I*5^(1/2)*sin(d*x+c)*cos(d*x+c)^2*(-2*(-
3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(I*(-1+cos(d*x+c))*5^
(1/2)/sin(d*x+c),1/5*I*5^(1/2))+3*I*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)*5^(1/2)*(-2*(-3+2*cos
(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))-3*I*5^(1/2)*sin(d
*x+c)*cos(d*x+c)^2*(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*Ellip
ticF(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))+5*I*5^(1/2)*sin(d*x+c)*cos(d*x+c)*(-2*(-3+2*cos(d*x+c
))/(1+cos(d*x+c)))^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x
+c),1/5*I*5^(1/2))-3*I*5^(1/2)*sin(d*x+c)*cos(d*x+c)*(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*2^(1/2)*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))+20*cos(d*x+c)^3-50*
cos(d*x+c)^2+30*cos(d*x+c))/cos(d*x+c)^(3/2)/sin(d*x+c)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) - 3)*cos(d*x + c)^(3/2)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2 \, \cos \left (d x + c\right ) - 3}{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(2*cos(d*x + c) - 3)*(cos(d*x + c) + 1)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^3 - 3*cos(d*x + c)^2),
x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (c + d x \right )} + 1}{\sqrt{2 \cos{\left (c + d x \right )} - 3} \cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+cos(d*x+c))/cos(d*x+c)**(3/2)/(-3+2*cos(d*x+c))**(1/2),x)
[Out]
Integral((cos(c + d*x) + 1)/(sqrt(2*cos(c + d*x) - 3)*cos(c + d*x)**(3/2)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) - 3)*cos(d*x + c)^(3/2)), x)