3.5.44 \(\int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) [444]

Optimal. Leaf size=95 \[ \frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{d} \]

[Out]

csc(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))*(-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 = 95, 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 {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[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 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 {-2-3 \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 {-2-3 \cos (c+d x)} (-\cos (c+d x))^{3/2}} \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (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 = 38.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \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[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(704\) vs. \(2(83)=166\).
time = 0.90, size = 705, normalized size = 7.42

method	result	size
default	\(\frac {\sqrt {-2-3 \cos \left (d x +c \right )}\, \left (2 \sqrt {10}\, \sqrt {\frac {2+3 \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}} \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 \sqrt {10}\, \sqrt {\frac {2+3 \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}} \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+2 \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {5}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sin \left (d x +c \right )-2 \sqrt {10}\, \sqrt {\frac {2+3 \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 {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \left (\cos ^{2}\left (d x +c \right )\right ) \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 )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2 \sqrt {10}\, \sqrt {\frac {2+3 \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 {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \cos \left (d x +c \right ) \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 )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {5}\, \cos \left (d x +c \right ) \sin \left (d x +c \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 )\right )}{10 d \left (2+3 \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {3}{2}} \sin \left (d x +c \right )}\)	\(705\)

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,method=_RETURNVERBOSE)

[Out]

1/10/d*(-2-3*cos(d*x+c))^(1/2)*(2*10^(1/2)*((2+3*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)*EllipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)^2*sin(d*x+c)+4*10^
(1/2)*((2+3*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)*EllipticF(1/5*5^(1/2)*
(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)*sin(d*x+c)+2*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*
5^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1
/2))*sin(d*x+c)-2*10^(1/2)*((2+3*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)*E
llipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)^2*sin(d*x+c)-10^(1/2)*((2+3*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*5^(1/2)*(-1+cos(d*x+c))/si
n(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)^2*sin(d*x+c)-2*10^(1/2)*((2+3*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*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)*
sin(d*x+c)-10^(1/2)*((2+3*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)*Elliptic
E(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*5^(1/2)*cos(d*x+c)*sin(d*x+c)+30*cos(d*x+c)^3-10*cos(d*x+c)^
2-20*cos(d*x+c))/(2+3*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)/(-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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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)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c) + 1)*sqrt(-3*cos(d*x + c) - 2)*sqrt(cos(d*x + c))/(3*cos(d*x + c)^3 + 2*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 {- 3 \cos {\left (c + d x \right )} - 2} \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)/(-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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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)/(-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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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 {-3\,\cos \left (c+d\,x\right )-2}} \,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)*(- 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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