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

Optimal. Leaf size=217 \[ \frac{4 a^3 (35 A+13 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (5 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}-\frac{4 a^3 (35 A-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}} \]

[Out]

(4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) -
(4*a^3*(35*A - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*
Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(3
5*A - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)

________________________________________________________________________________________

Rubi [A]  time = 0.587426, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3044, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{4 a^3 (35 A+13 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (5 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}-\frac{4 a^3 (35 A-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) -
(4*a^3*(35*A - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*
Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(3
5*A - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)

Rule 3044

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[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[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 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 \frac{(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 \int \frac{(a+a \cos (c+d x))^3 \left (3 a A-\frac{1}{2} a (7 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}+\frac{4 \int \frac{(a+a \cos (c+d x))^2 \left (\frac{1}{4} a^2 (35 A+C)-\frac{1}{4} a^2 (35 A-11 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{7 a}\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}-\frac{2 (35 A-11 C) \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{8 \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{2} a^3 (35 A+4 C)-\frac{1}{4} a^3 (35 A-41 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}-\frac{2 (35 A-11 C) \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{8 \int \frac{\frac{1}{2} a^4 (35 A+4 C)+\left (-\frac{1}{4} a^4 (35 A-41 C)+\frac{1}{2} a^4 (35 A+4 C)\right ) \cos (c+d x)-\frac{1}{4} a^4 (35 A-41 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=-\frac{4 a^3 (35 A-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}-\frac{2 (35 A-11 C) \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{16 \int \frac{\frac{5}{8} a^4 (35 A+13 C)+\frac{21}{8} a^4 (5 A+7 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{105 a}\\ &=-\frac{4 a^3 (35 A-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}-\frac{2 (35 A-11 C) \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (2 a^3 (35 A+13 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (5 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^3 (35 A+13 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^3 (35 A-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (7 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d}-\frac{2 (35 A-11 C) \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d}\\ \end{align*}

Mathematica [C]  time = 6.48084, size = 926, normalized size = 4.27 \[ \sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left (-\frac{(15 \cos (2 c) A+5 A+14 C+14 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{4 d}+\frac{(28 A+107 C) \cos (d x) \sin (c)}{336 d}+\frac{3 C \cos (2 d x) \sin (2 c)}{40 d}+\frac{C \cos (3 d x) \sin (3 c)}{112 d}+\frac{(28 A+107 C) \cos (c) \sin (d x)}{336 d}+\frac{3 C \cos (2 c) \sin (2 d x)}{40 d}+\frac{C \cos (3 c) \sin (3 d x)}{112 d}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )-\frac{A (\cos (c+d x) a+a)^3 \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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d}-\frac{7 C (\cos (c+d x) a+a)^3 \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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{6 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x) a+a)^3 \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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{42 d \sqrt{\cot ^2(c)+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-((5*A + 14*C + 15*A*Cos[2*c] + 14*C*Cos[2*c])
*Csc[c]*Sec[c])/(40*d) + ((28*A + 107*C)*Cos[d*x]*Sin[c])/(336*d) + (3*C*Cos[2*d*x]*Sin[2*c])/(40*d) + (C*Cos[
3*d*x]*Sin[3*c])/(112*d) + ((28*A + 107*C)*Cos[c]*Sin[d*x])/(336*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) +
(3*C*Cos[2*c]*Sin[2*d*x])/(40*d) + (C*Cos[3*c]*Sin[3*d*x])/(112*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*Hype
rgeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4,
1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - Ar
cTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]
])/(42*d*Sqrt[1 + Cot[c]^2]) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((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[Tan[
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 + T
an[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]]]*Sqr
t[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (7
*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTa
n[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[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]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Si
n[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)

________________________________________________________________________________________

Maple [B]  time = 0.106, size = 569, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/105*a^3*(120*C*(-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)^8
-432*C*(-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)^6+14*(-2*sin
(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(5*A+43*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-4*(-2*sin(1/2
*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(35*A+52*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-105*A*(-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)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2))+175*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-147*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)*EllipticE(cos(1/2
*d*x+1/2*c),2^(1/2))+65*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)*EllipticF(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)/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} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3/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 (\frac{C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}}{\cos \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*a^3*cos(d*x + c)^5 + 3*C*a^3*cos(d*x + c)^4 + (A + 3*C)*a^3*cos(d*x + c)^3 + (3*A + C)*a^3*cos(d*x
+ c)^2 + 3*A*a^3*cos(d*x + c) + A*a^3)/cos(d*x + c)^(3/2), 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+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(3/2),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} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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