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

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

[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 = 0.19, 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 = {2995, 2994} \[ \frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[c + d*x])/(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 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]

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]

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 = 30.16, size = 0, normalized size = 0.00 \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \]

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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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt {-3 \, \cos \left (d x + c\right ) - 2} \sqrt {\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2}}, x\right ) \]

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(-(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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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 = 0.39, size = 703, normalized size = 7.40 \[ -\frac {\sqrt {-2-3 \cos \left (d x +c \right )}\, \left (-2 \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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}-4 \sqrt {2}\, \sin \left (d x +c \right ) \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 )}}\, \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}+\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 ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {5}+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 ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {5}-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 )+\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 ) \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {5}+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 ) \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {5}-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 )} \]

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)*sin(d*x+c)*cos(d*x+c)^2*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))
^(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)-4*2
^(1/2)*sin(d*x+c)*cos(d*x+c)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(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)+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))/sin(d*x+c),5^(1/2))*sin
(d*x+c)*cos(d*x+c)^2*5^(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)))^(1/2)*EllipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*sin(d*x+c)*cos(d*x+c)^2*5^(1/2)-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)+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))/sin(d*x+c),5^(1/2))*sin(d*x+c)*cos(d*x+
c)*5^(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)))^(1/2)*Ellipt
icF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*sin(d*x+c)*cos(d*x+c)*5^(1/2)-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 \[ \int \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

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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