3.134 \(\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=230 \[ \frac{8 a^2 (33 A+25 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (33 A+25 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d} \]

[Out]

(4*a^2*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF[(c + d*x)/2, 2])/(231*d)
+ (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c
+ d*x])/(45*d) + (2*a^2*(99*A + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a +
a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d
)

________________________________________________________________________________________

Rubi [A]  time = 0.478117, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639} \[ \frac{8 a^2 (33 A+25 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (33 A+25 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]

[Out]

(4*a^2*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF[(c + d*x)/2, 2])/(231*d)
+ (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c
+ d*x])/(45*d) + (2*a^2*(99*A + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a +
a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d
)

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*
sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -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 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]

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]

Rubi steps

\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{2 \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac{1}{2} a (11 A+5 C)+2 a C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}+\frac{4 \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (99 A+65 C)+\frac{1}{4} a^2 (99 A+89 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}+\frac{4 \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{4} a^3 (99 A+65 C)+\left (\frac{1}{4} a^3 (99 A+65 C)+\frac{1}{4} a^3 (99 A+89 C)\right ) \cos (c+d x)+\frac{1}{4} a^3 (99 A+89 C) \cos ^2(c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 a^2 (99 A+89 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}+\frac{8 \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{2} a^3 (33 A+25 C)+\frac{77}{4} a^3 (9 A+7 C) \cos (c+d x)\right ) \, dx}{693 a}\\ &=\frac{2 a^2 (99 A+89 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}+\frac{1}{9} \left (2 a^2 (9 A+7 C)\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (4 a^2 (33 A+25 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a^2 (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a^2 (99 A+89 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}+\frac{1}{15} \left (2 a^2 (9 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (4 a^2 (33 A+25 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{8 a^2 (33 A+25 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a^2 (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a^2 (99 A+89 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d}\\ \end{align*}

Mathematica [C]  time = 6.28686, size = 982, normalized size = 4.27 \[ \sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (-\frac{(9 A+7 C) \cot (c)}{15 d}+\frac{(1122 A+941 C) \cos (d x) \sin (c)}{3696 d}+\frac{(18 A+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(44 A+101 C) \cos (3 d x) \sin (3 c)}{2464 d}+\frac{C \cos (4 d x) \sin (4 c)}{72 d}+\frac{C \cos (5 d x) \sin (5 c)}{352 d}+\frac{(1122 A+941 C) \cos (c) \sin (d x)}{3696 d}+\frac{(18 A+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(44 A+101 C) \cos (3 c) \sin (3 d x)}{2464 d}+\frac{C \cos (4 c) \sin (4 d x)}{72 d}+\frac{C \cos (5 c) \sin (5 d x)}{352 d}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )-\frac{3 A (\cos (c+d x) a+a)^2 \csc (c) \left (\frac{\, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt{\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}}}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{10 d}-\frac{7 C (\cos (c+d x) a+a)^2 \csc (c) \left (\frac{\, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt{\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}}}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{30 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{7 d \sqrt{\cot ^2(c)+1}}-\frac{50 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{231 d \sqrt{\cot ^2(c)+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]

[Out]

Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-((9*A + 7*C)*Cot[c])/(15*d) + ((1122*A + 941*
C)*Cos[d*x]*Sin[c])/(3696*d) + ((18*A + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 101*C)*Cos[3*d*x]*Sin[3*
c])/(2464*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(352*d) + ((1122*A + 941*C)*Cos[c]*Sin
[d*x])/(3696*d) + ((18*A + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((44*A + 101*C)*Cos[3*c]*Sin[3*d*x])/(2464*d)
+ (C*Cos[4*c]*Sin[4*d*x])/(72*d) + (C*Cos[5*c]*Sin[5*d*x])/(352*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*Hyper
geometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sq
rt[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*Sqrt[1 + Cot[c]^2]) - (50*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1
/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - Arc
Tan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]]
)/(231*d*Sqrt[1 + Cot[c]^2]) - (3*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1
/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Ta
n[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 +
Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*S
qrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) -
(7*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + Ar
cTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + Arc
Tan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcT
an[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 +
Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d)

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Maple [A]  time = 0.083, size = 436, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2),x)

[Out]

-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*(10080*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12-37520*C*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(3960*A+57040*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2
*c)+(-11484*A-46192*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(12474*A+22022*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*
d*x+1/2*c)+(-3861*A-4563*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+990*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(
1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2079*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2
*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+750*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1617*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)*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)^(1/2)/si
n(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{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^2*cos(d*x + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{2} \cos \left (d x + c\right )^{5} + 2 \, C a^{2} \cos \left (d x + c\right )^{4} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, A a^{2} \cos \left (d x + c\right )^{2} + A a^{2} \cos \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*a^2*cos(d*x + c)^5 + 2*C*a^2*cos(d*x + c)^4 + (A + C)*a^2*cos(d*x + c)^3 + 2*A*a^2*cos(d*x + c)^2
+ A*a^2*cos(d*x + c))*sqrt(cos(d*x + c)), 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)**(3/2)*(a+a*cos(d*x+c))**2*(A+C*cos(d*x+c)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^2*cos(d*x + c)^(3/2), x)