3.456 \(\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{a+a \cos (c+d x)} \, dx\)
Optimal. Leaf size=210 \[ \frac{5 (7 A-7 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A-7 B+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-7 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A-7 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d} \]
[Out]
(-3*(5*A - 7*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A - 7*B + 9*C)*EllipticF[(c + d*x)/2, 2])/(21
*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A - 7*B + 7*C)*Cos[c + d*x]^(3/2)
*Sin[c + d*x])/(5*a*d) + ((7*A - 7*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*
x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))
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Rubi [A] time = 0.252536, antiderivative size = 210, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116, Rules used =
{3041, 2748, 2635, 2639, 2641} \[ \frac{5 (7 A-7 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A-7 B+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-7 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A-7 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]
[Out]
(-3*(5*A - 7*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A - 7*B + 9*C)*EllipticF[(c + d*x)/2, 2])/(21
*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A - 7*B + 7*C)*Cos[c + d*x]^(3/2)
*Sin[c + d*x])/(5*a*d) + ((7*A - 7*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*
x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))
Rule 3041
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((a*A - b*B + 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)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(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, B, 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 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[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{\cos ^{\frac{5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx &=-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^{\frac{5}{2}}(c+d x) \left (-\frac{1}{2} a (5 A-7 B+7 C)+\frac{1}{2} a (7 A-7 B+9 C) \cos (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(5 A-7 B+7 C) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{(7 A-7 B+9 C) \int \cos ^{\frac{7}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac{(5 A-7 B+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(7 A-7 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 (5 A-7 B+7 C)) \int \sqrt{\cos (c+d x)} \, dx}{10 a}+\frac{(5 (7 A-7 B+9 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{14 a}\\ &=-\frac{3 (5 A-7 B+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 a d}-\frac{(5 A-7 B+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(7 A-7 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{(5 (7 A-7 B+9 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac{3 (5 A-7 B+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{5 (7 A-7 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 a d}-\frac{(5 A-7 B+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(7 A-7 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.85507, size = 1752, normalized size = 8.34 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]
[Out]
(((-3*I)/4)*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)]*Sq
rt[(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*C
os[c + d*x]) + (((21*I)/20)*B*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]) - (((21*I)/20)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hyperge
ometric2F1[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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((2*(5*A - 5*B + 5*C +
10*A*Cos[c] - 16*B*Cos[c] + 16*C*Cos[c])*Csc[c])/(5*d) + ((28*A - 28*B + 51*C)*Cos[d*x]*Sin[c])/(21*d) + (2*(
B - C)*Cos[2*d*x]*Sin[2*c])/(5*d) + (C*Cos[3*d*x]*Sin[3*c])/(7*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x
)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + ((28*A - 28*B + 51*C)*Cos[c]*Sin[d*x])/(21*d) + (2*(B - C)*Cos[2*
c]*Sin[2*d*x])/(5*d) + (C*Cos[3*c]*Sin[3*d*x])/(7*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 - ArcTan[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]) + (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Hypergeom
etricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[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]) - (15*C*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 - ArcTan[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]]]])/(7*d*
(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])
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Maple [A] time = 0.313, size = 341, normalized size = 1.6 \begin{align*} -{\frac{1}{105\,ad}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 175\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +315\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -175\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -441\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +225\,C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +441\,C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) -480\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 336\,B+864\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -280\,A-392\,B-888\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 630\,A-210\,B+930\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -245\,A+161\,B-321\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x)
[Out]
-1/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(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)*(175*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+315*A*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))-175*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-441*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+225*C*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))+441*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-480*C*sin(1/2*d*x+1/2*c)^10+(
336*B+864*C)*sin(1/2*d*x+1/2*c)^8+(-280*A-392*B-888*C)*sin(1/2*d*x+1/2*c)^6+(630*A-210*B+930*C)*sin(1/2*d*x+1/
2*c)^4+(-245*A+161*B-321*C)*sin(1/2*d*x+1/2*c)^2)/a/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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}{a \cos \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(5/2)/(a*cos(d*x + c) + a), x)
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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 )^{4} + B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}}{a \cos \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="fricas")
[Out]
integral((C*cos(d*x + c)^4 + B*cos(d*x + c)^3 + A*cos(d*x + c)^2)*sqrt(cos(d*x + c))/(a*cos(d*x + c) + a), 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(cos(d*x+c)**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}{a \cos \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(5/2)/(a*cos(d*x + c) + a), x)