3.66 \(\int \frac{\cos ^3(c+d x) (A+C \cos ^2(c+d x))}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=174 \[ \frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[Out]

(-4*C*x)/a^4 + (2*(3*A + 122*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^
4*d*(1 + Cos[c + d*x])^2) + (4*C*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*
x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)

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Rubi [A]  time = 0.577124, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3042, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]

[Out]

(-4*C*x)/a^4 + (2*(3*A + 122*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^
4*d*(1 + Cos[c + d*x])^2) + (4*C*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*
x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)

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 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{\cos ^3(c+d x) (a (3 A-4 C)+a (A+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos ^2(c+d x) \left (6 a^2 (A-6 C)+a^2 (3 A+52 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos (c+d x) \left (2 a^3 (3 A-88 C)+2 a^3 (3 A+122 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{2 a^3 (3 A-88 C) \cos (c+d x)+2 a^3 (3 A+122 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int -\frac{420 a^4 C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7}\\ &=\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(4 C) \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac{4 C x}{a^4}+\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(4 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac{4 C x}{a^4}+\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{4 C \sin (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.729315, size = 371, normalized size = 2.13 \[ -\frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (2520 A \sin \left (c+\frac{d x}{2}\right )-1764 A \sin \left (c+\frac{3 d x}{2}\right )+1260 A \sin \left (2 c+\frac{3 d x}{2}\right )-588 A \sin \left (2 c+\frac{5 d x}{2}\right )+420 A \sin \left (3 c+\frac{5 d x}{2}\right )-144 A \sin \left (3 c+\frac{7 d x}{2}\right )-2520 A \sin \left (\frac{d x}{2}\right )+46130 C \sin \left (c+\frac{d x}{2}\right )-46116 C \sin \left (c+\frac{3 d x}{2}\right )+18060 C \sin \left (2 c+\frac{3 d x}{2}\right )-19292 C \sin \left (2 c+\frac{5 d x}{2}\right )+2100 C \sin \left (3 c+\frac{5 d x}{2}\right )-3791 C \sin \left (3 c+\frac{7 d x}{2}\right )-735 C \sin \left (4 c+\frac{7 d x}{2}\right )-105 C \sin \left (4 c+\frac{9 d x}{2}\right )-105 C \sin \left (5 c+\frac{9 d x}{2}\right )+29400 C d x \cos \left (c+\frac{d x}{2}\right )+17640 C d x \cos \left (c+\frac{3 d x}{2}\right )+17640 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+5880 C d x \cos \left (2 c+\frac{5 d x}{2}\right )+5880 C d x \cos \left (3 c+\frac{5 d x}{2}\right )+840 C d x \cos \left (3 c+\frac{7 d x}{2}\right )+840 C d x \cos \left (4 c+\frac{7 d x}{2}\right )-60830 C \sin \left (\frac{d x}{2}\right )+29400 C d x \cos \left (\frac{d x}{2}\right )\right )}{26880 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]

[Out]

-(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*C*d*x*Cos[(d*x)/2] + 29400*C*d*x*Cos[c + (d*x)/2] + 17640*C*d*x*Cos[c + (
3*d*x)/2] + 17640*C*d*x*Cos[2*c + (3*d*x)/2] + 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5*d*x)/
2] + 840*C*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 2520*A*Sin[(d*x)/2] - 60830*C*Sin[(d*x)
/2] + 2520*A*Sin[c + (d*x)/2] + 46130*C*Sin[c + (d*x)/2] - 1764*A*Sin[c + (3*d*x)/2] - 46116*C*Sin[c + (3*d*x)
/2] + 1260*A*Sin[2*c + (3*d*x)/2] + 18060*C*Sin[2*c + (3*d*x)/2] - 588*A*Sin[2*c + (5*d*x)/2] - 19292*C*Sin[2*
c + (5*d*x)/2] + 420*A*Sin[3*c + (5*d*x)/2] + 2100*C*Sin[3*c + (5*d*x)/2] - 144*A*Sin[3*c + (7*d*x)/2] - 3791*
C*Sin[3*c + (7*d*x)/2] - 735*C*Sin[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (9*d*x)/2] - 105*C*Sin[5*c + (9*d*x)/2])
)/(26880*a^4*d)

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Maple [A]  time = 0.031, size = 210, normalized size = 1.2 \begin{align*} -{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{23\,C}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}-8\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*A-1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*C+3/40/d/a^4*A*tan(1/2*d*x+1/2*c)^5+7/40/d/
a^4*C*tan(1/2*d*x+1/2*c)^5-1/8/d/a^4*tan(1/2*d*x+1/2*c)^3*A-23/24/d/a^4*C*tan(1/2*d*x+1/2*c)^3+1/8/d/a^4*A*tan
(1/2*d*x+1/2*c)+49/8/d/a^4*C*tan(1/2*d*x+1/2*c)+2/d/a^4*C*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)-8/d/a^4*
arctan(tan(1/2*d*x+1/2*c))*C

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Maxima [A]  time = 1.65982, size = 332, normalized size = 1.91 \begin{align*} \frac{C{\left (\frac{1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac{a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{6720 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + \frac{3 \, A{\left (\frac{35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(C*(1680*sin(d*x + c)/((a^4 + a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d
*x + c)/(cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
- 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) + 3*A*(35*s
in(d*x + c)/(cos(d*x + c) + 1) - 35*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)
^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4)/d

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Fricas [A]  time = 1.41748, size = 516, normalized size = 2.97 \begin{align*} -\frac{420 \, C d x \cos \left (d x + c\right )^{4} + 1680 \, C d x \cos \left (d x + c\right )^{3} + 2520 \, C d x \cos \left (d x + c\right )^{2} + 1680 \, C d x \cos \left (d x + c\right ) + 420 \, C d x -{\left (105 \, C \cos \left (d x + c\right )^{4} + 4 \,{\left (9 \, A + 296 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (39 \, A + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (6 \, A + 559 \, C\right )} \cos \left (d x + c\right ) + 6 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/105*(420*C*d*x*cos(d*x + c)^4 + 1680*C*d*x*cos(d*x + c)^3 + 2520*C*d*x*cos(d*x + c)^2 + 1680*C*d*x*cos(d*x
+ c) + 420*C*d*x - (105*C*cos(d*x + c)^4 + 4*(9*A + 296*C)*cos(d*x + c)^3 + (39*A + 2636*C)*cos(d*x + c)^2 + 4
*(6*A + 559*C)*cos(d*x + c) + 6*A + 664*C)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^
4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)

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Sympy [A]  time = 40.2136, size = 462, normalized size = 2.66 \begin{align*} \begin{cases} - \frac{15 A \tan ^{9}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} + \frac{48 A \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} - \frac{42 A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} + \frac{105 A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} - \frac{3360 C d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} - \frac{3360 C d x}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} - \frac{15 C \tan ^{9}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} + \frac{132 C \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} - \frac{658 C \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} + \frac{4340 C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} + \frac{6825 C \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 840 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{3}{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((-15*A*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 48*A*tan(c/2 + d*x/2)**7/
(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 42*A*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840
*a**4*d) + 105*A*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*C*d*x*tan(c/2 + d*x/2)*
*2/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*C*d*x/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) -
15*C*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 132*C*tan(c/2 + d*x/2)**7/(840*a**4*d
*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 658*C*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) +
4340*C*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825*C*tan(c/2 + d*x/2)/(840*a**4*
d*tan(c/2 + d*x/2)**2 + 840*a**4*d), Ne(d, 0)), (x*(A + C*cos(c)**2)*cos(c)**3/(a*cos(c) + a)**4, True))

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Giac [A]  time = 1.16189, size = 248, normalized size = 1.43 \begin{align*} -\frac{\frac{3360 \,{\left (d x + c\right )} C}{a^{4}} - \frac{1680 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 63 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/840*(3360*(d*x + c)*C/a^4 - 1680*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^4) + (15*A*a^24*tan
(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 63*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 147*C*a^24*tan(1/2
*d*x + 1/2*c)^5 + 105*A*a^24*tan(1/2*d*x + 1/2*c)^3 + 805*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 105*A*a^24*tan(1/2*d
*x + 1/2*c) - 5145*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d