3.5.48 \(\int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) [448]

Optimal. Leaf size=96 \[ -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 = {3074, 3073} \begin {gather*} -\frac {2 \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 (\left .\text {ArcSin}\left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3073

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]/(f*b*c^2))*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)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3074

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]

Rubi steps

\begin {align*} \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx &=-\frac {\sqrt {-\cos (c+d x)} \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} (-\cos (c+d x))^{3/2}} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-3-2 \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)}}{3 d}\\ \end {align*}

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Mathematica [F]
time = 38.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (85 ) = 170\).
time = 0.56, size = 740, normalized size = 7.71

method	result	size
default	\(-\frac {\sqrt {-3-2 \cos \left (d x +c \right )}\, \left (3 i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 {5}+6 i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 {5}-3 i \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {5}+3 i \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 {5}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right )+i \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {5}-3 i \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {5}+i \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {5}-20 \left (\cos ^{3}\left (d x +c \right )\right )-10 \left (\cos ^{2}\left (d x +c \right )\right )+30 \cos \left (d x +c \right )\right )}{15 d \left (3+2 \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {3}{2}} \sin \left (d x +c \right )}\)	\(740\)

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,method=_RETURNVERBOSE)

[Out]

-1/15/d*(-3-2*cos(d*x+c))^(1/2)*(3*I*EllipticF(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/2))*cos(d*x+c)^
2*10^(1/2)*((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)))^(3/2)*sin(d*x+c)*5^(1/2
)+6*I*EllipticF(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/2))*cos(d*x+c)*10^(1/2)*((3+2*cos(d*x+c))/(1+c
os(d*x+c)))^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)*5^(1/2)-3*I*10^(1/2)*((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(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*
x+c),I*5^(1/2))*cos(d*x+c)^2*sin(d*x+c)*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)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/
2))+I*10^(1/2)*((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(1/5
*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/2))*cos(d*x+c)^2*sin(d*x+c)*5^(1/2)-3*I*10^(1/2)*((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(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(
d*x+c),I*5^(1/2))*cos(d*x+c)*sin(d*x+c)*5^(1/2)+I*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*2^(1/2)*(co
s(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/2))*cos(d*x+c)*sin(d*
x+c)*5^(1/2)-20*cos(d*x+c)^3-10*cos(d*x+c)^2+30*cos(d*x+c))/(3+2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3-2*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c) + 1)*sqrt(-2*cos(d*x + c) - 3)*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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