3.441 \(\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {2+3 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=72 \[ -\frac {\cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right )}{d} \]

[Out]

-cot(d*x+c)*EllipticE(1/5*(2+3*cos(d*x+c))^(1/2)*5^(1/2)/cos(d*x+c)^(1/2),5^(1/2))*(-1-sec(d*x+c))^(1/2)*(1-se
c(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2994} \[ -\frac {\cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]),x]

[Out]

-((Cot[c + d*x]*EllipticE[ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c +
d*x]]*Sqrt[1 - Sec[c + d*x]])/d)

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 {2+3 \cos (c+d x)}} \, dx &=-\frac {\cot (c+d x) E\left (\left .\sin ^{-1}\left (\frac {\sqrt {2+3 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{d}\\ \end {align*}

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Mathematica [F]  time = 35.24, size = 0, normalized size = 0.00 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {2+3 \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]),x]

[Out]

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]), x]

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, \cos \left (d x + c\right ) + 2} {\left (\cos \left (d x + c\right ) + 1\right )} \sqrt {\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(3*cos(d*x + c) + 2)*(cos(d*x + c) + 1)*sqrt(cos(d*x + c))/(3*cos(d*x + c)^3 + 2*cos(d*x + c)^2),
x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) + 2} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(3*cos(d*x + c) + 2)*cos(d*x + c)^(3/2)), x)

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maple [B]  time = 0.45, size = 658, normalized size = 9.14 \[ -\frac {2 \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )+4 \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )+2 \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )-5 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )+2 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )-5 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )+2 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )+30 \left (\cos ^{3}\left (d x +c \right )\right )-10 \left (\cos ^{2}\left (d x +c \right )\right )-20 \cos \left (d x +c \right )}{10 d \sqrt {2+3 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \sin \left (d x +c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10/d/(2+3*cos(d*x+c))^(1/2)*(2*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(
d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))+4*2^(1/2)*(cos(d*x+c)
/(1+cos(d*x+c)))^(3/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)*EllipticF((-1+co
s(d*x+c))/sin(d*x+c),1/5*5^(1/2))+2*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)*10^(1/2)*((2+3*cos(d*
x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))-5*sin(d*x+c)*cos(d*x+c)^2*2^(1/2
)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))
/sin(d*x+c),1/5*5^(1/2))+2*sin(d*x+c)*cos(d*x+c)^2*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*co
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))-5*sin(d*x+c)*cos(d*x+c)*2^(1
/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c
))/sin(d*x+c),1/5*5^(1/2))+2*sin(d*x+c)*cos(d*x+c)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*co
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))+30*cos(d*x+c)^3-10*cos(d*x+c
)^2-20*cos(d*x+c))/cos(d*x+c)^(3/2)/sin(d*x+c)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) + 2} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(3*cos(d*x + c) + 2)*cos(d*x + c)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {3 \cos {\left (c + d x \right )} + 2} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cos(c + d*x) + 1)/(sqrt(3*cos(c + d*x) + 2)*cos(c + d*x)**(3/2)), x)

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