3.157 \(\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx\)
Optimal. Leaf size=192 \[ -\frac{(5 A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}} \]
[Out]
(-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A +
5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7
*A + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c
+ d*x]))
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Rubi [A] time = 0.211847, antiderivative size = 192, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{3042, 2748, 2636, 2639, 2641} \[ -\frac{(5 A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]
[Out]
(-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A +
5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7
*A + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c
+ d*x]))
Rule 3042
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x
])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)
*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2
) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Rule 2636
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]
Rule 2639
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx &=-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))}+\frac{\int \frac{\frac{1}{2} a (7 A+5 C)-\frac{1}{2} a (5 A+3 C) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))}-\frac{(5 A+3 C) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{2 a}+\frac{(7 A+5 C) \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{2 a}\\ &=\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))}-\frac{(5 A+3 C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}+\frac{(3 (7 A+5 C)) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{10 a}\\ &=-\frac{(5 A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))}-\frac{(3 (7 A+5 C)) \int \sqrt{\cos (c+d x)} \, dx}{10 a}\\ &=-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(5 A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 7.09504, size = 1207, normalized size = 6.29 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]
[Out]
(((-21*I)/20)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^
((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^
(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*
d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*
Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[
2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a
*Cos[c + d*x]) - (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2,
3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*
x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x
))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] +
I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((
2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Si
n[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((16*A + 10*C + 5*A*Cos[c] + 5*C*Cos[
c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4
*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] - 5*A*Sin[d*x]))/(15*d) - (4*S
ec[c]*Sec[c + d*x]*(5*A*Sin[c] - 24*A*Sin[d*x] - 15*C*Sin[d*x]))/(15*d)))/(a + a*Cos[c + d*x]) + (5*A*Cos[c/2
+ (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - Arc
Tan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*
Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (C*Cos[c/2 + (d*x)/2]^2*C
sc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*S
qrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d
*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])
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Maple [B] time = 0.365, size = 803, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c)),x)
[Out]
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/a*(-2/5*A/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/
2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c
)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1
/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))*(-2*sin(1/2*d*x+
1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-2*A*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*
sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+(2*A+2*C)*(-(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin
(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)+(-A-C)*(cos(1/2*d*x+1/2*c)*(2*sin(1/2*d*x+1
/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2
*c),2^(1/2)))-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)/cos(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2
*d*x+1/2*c)^2)^(1/2))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c)),x, algorithm="maxima")
[Out]
Timed out
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{\cos \left (d x + c\right )}}{a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c)),x, algorithm="fricas")
[Out]
integral((C*cos(d*x + c)^2 + A)*sqrt(cos(d*x + c))/(a*cos(d*x + c)^5 + a*cos(d*x + c)^4), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)**2)/cos(d*x+c)**(7/2)/(a+a*cos(d*x+c)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c)),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + A)/((a*cos(d*x + c) + a)*cos(d*x + c)^(7/2)), x)