3.73 \(\int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=150 \[ \frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {4 \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {4 x}{a^4}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {12 \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[Out]

-4*x/a^4+244/105*sin(d*x+c)/a^4/d-88/105*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2+4*sin(d*x+c)/a^4/d/(1+
cos(d*x+c))-1/7*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^4-12/35*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c)
)^3

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Rubi [A]  time = 0.37, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2765, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {4 \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {4 x}{a^4}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {12 \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^4,x]

[Out]

(-4*x)/a^4 + (244*Sin[c + d*x])/(105*a^4*d) - (88*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2
) + (4*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) -
(12*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)

Rule 12

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

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]

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 2765

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

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {\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) (4 a-8 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \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 (36 a^2-52 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \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 (176 a^3-244 a^3 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {\int \frac {176 a^3 \cos (c+d x)-244 a^3 \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \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 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7}\\ &=\frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {4 \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac {4 x}{a^4}+\frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {4 \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac {4 x}{a^4}+\frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {4 \sin (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 263, normalized size = 1.75 \[ -\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (46130 \sin \left (c+\frac {d x}{2}\right )-46116 \sin \left (c+\frac {3 d x}{2}\right )+18060 \sin \left (2 c+\frac {3 d x}{2}\right )-19292 \sin \left (2 c+\frac {5 d x}{2}\right )+2100 \sin \left (3 c+\frac {5 d x}{2}\right )-3791 \sin \left (3 c+\frac {7 d x}{2}\right )-735 \sin \left (4 c+\frac {7 d x}{2}\right )-105 \sin \left (4 c+\frac {9 d x}{2}\right )-105 \sin \left (5 c+\frac {9 d x}{2}\right )+29400 d x \cos \left (c+\frac {d x}{2}\right )+17640 d x \cos \left (c+\frac {3 d x}{2}\right )+17640 d x \cos \left (2 c+\frac {3 d x}{2}\right )+5880 d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 d x \cos \left (3 c+\frac {5 d x}{2}\right )+840 d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 d x \cos \left (4 c+\frac {7 d x}{2}\right )-60830 \sin \left (\frac {d x}{2}\right )+29400 d x \cos \left (\frac {d x}{2}\right )\right )}{26880 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^4,x]

[Out]

-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*d*x*Cos[(d*x)/2] + 29400*d*x*Cos[c + (d*x)/2] + 17640*d*x*Cos[c +
(3*d*x)/2] + 17640*d*x*Cos[2*c + (3*d*x)/2] + 5880*d*x*Cos[2*c + (5*d*x)/2] + 5880*d*x*Cos[3*c + (5*d*x)/2] +
840*d*x*Cos[3*c + (7*d*x)/2] + 840*d*x*Cos[4*c + (7*d*x)/2] - 60830*Sin[(d*x)/2] + 46130*Sin[c + (d*x)/2] - 4
6116*Sin[c + (3*d*x)/2] + 18060*Sin[2*c + (3*d*x)/2] - 19292*Sin[2*c + (5*d*x)/2] + 2100*Sin[3*c + (5*d*x)/2]
- 3791*Sin[3*c + (7*d*x)/2] - 735*Sin[4*c + (7*d*x)/2] - 105*Sin[4*c + (9*d*x)/2] - 105*Sin[5*c + (9*d*x)/2]))
/(a^4*d)

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fricas [A]  time = 0.65, size = 162, normalized size = 1.08 \[ -\frac {420 \, d x \cos \left (d x + c\right )^{4} + 1680 \, d x \cos \left (d x + c\right )^{3} + 2520 \, d x \cos \left (d x + c\right )^{2} + 1680 \, d x \cos \left (d x + c\right ) + 420 \, d x - {\left (105 \, \cos \left (d x + c\right )^{4} + 1184 \, \cos \left (d x + c\right )^{3} + 2636 \, \cos \left (d x + c\right )^{2} + 2236 \, \cos \left (d x + c\right ) + 664\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/105*(420*d*x*cos(d*x + c)^4 + 1680*d*x*cos(d*x + c)^3 + 2520*d*x*cos(d*x + c)^2 + 1680*d*x*cos(d*x + c) + 4
20*d*x - (105*cos(d*x + c)^4 + 1184*cos(d*x + c)^3 + 2636*cos(d*x + c)^2 + 2236*cos(d*x + c) + 664)*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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giac [A]  time = 0.49, size = 112, normalized size = 0.75 \[ -\frac {\frac {3360 \, {\left (d x + c\right )}}{a^{4}} - \frac {1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 126, normalized size = 0.84 \[ -\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d \,a^{4}}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*cos(d*x+c))^4,x)

[Out]

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

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maxima [A]  time = 0.81, size = 158, normalized size = 1.05 \[ \frac {\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(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)/d

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mupad [B]  time = 0.47, size = 137, normalized size = 0.91 \[ -\frac {15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-192\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+1144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-6112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )}{840\,a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + a*cos(c + d*x))^4,x)

[Out]

-(15*sin(c/2 + (d*x)/2) - 192*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 1144*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2) - 6112*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) - 1680*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 3360*co
s(c/2 + (d*x)/2)^7*(c + d*x))/(840*a^4*d*cos(c/2 + (d*x)/2)^7)

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sympy [A]  time = 19.07, size = 280, normalized size = 1.87 \[ \begin {cases} - \frac {3360 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+a*cos(d*x+c))**4,x)

[Out]

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

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