3.91 \(\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=168 \[ \frac{496 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{5 \tan (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{67 \tan (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \tan (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{5 \tan (c+d x)}{21 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

[Out]

(-5*ArcTanh[Sin[c + d*x]])/(a^5*d) + (496*Tan[c + d*x])/(63*a^5*d) - Tan[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5)
- (5*Tan[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Tan[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (6
7*Tan[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) - (5*Tan[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))

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Rubi [A]  time = 0.532824, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2766, 2978, 2748, 3767, 8, 3770} \[ \frac{496 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{5 \tan (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{67 \tan (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \tan (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{5 \tan (c+d x)}{21 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]

[Out]

(-5*ArcTanh[Sin[c + d*x]])/(a^5*d) + (496*Tan[c + d*x])/(63*a^5*d) - Tan[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5)
- (5*Tan[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Tan[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (6
7*Tan[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) - (5*Tan[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))

Rule 2766

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

Rule 2978

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*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[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] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])

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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{\int \frac{(10 a-5 a \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}+\frac{\int \frac{\left (85 a^2-60 a^2 \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (570 a^3-435 a^3 \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \tan (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (2715 a^4-2010 a^4 \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \tan (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}+\frac{\int \left (7440 a^5-4725 a^5 \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{945 a^{10}}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \tan (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}-\frac{5 \int \sec (c+d x) \, dx}{a^5}+\frac{496 \int \sec ^2(c+d x) \, dx}{63 a^5}\\ &=-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \tan (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}-\frac{496 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{63 a^5 d}\\ &=-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{496 \tan (c+d x)}{63 a^5 d}-\frac{\tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \tan (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \tan (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \tan (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 6.33967, size = 453, normalized size = 2.7 \[ \frac{160 \cos ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}-\frac{160 \cos ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}+\frac{\sec \left (\frac{c}{2}\right ) \sec (c) \left (-56952 \sin \left (c-\frac{d x}{2}\right )+43722 \sin \left (c+\frac{d x}{2}\right )-47208 \sin \left (2 c+\frac{d x}{2}\right )-18144 \sin \left (c+\frac{3 d x}{2}\right )+41796 \sin \left (2 c+\frac{3 d x}{2}\right )-28350 \sin \left (3 c+\frac{3 d x}{2}\right )+34578 \sin \left (c+\frac{5 d x}{2}\right )-5691 \sin \left (2 c+\frac{5 d x}{2}\right )+28719 \sin \left (3 c+\frac{5 d x}{2}\right )-11550 \sin \left (4 c+\frac{5 d x}{2}\right )+15517 \sin \left (2 c+\frac{7 d x}{2}\right )-504 \sin \left (3 c+\frac{7 d x}{2}\right )+13186 \sin \left (4 c+\frac{7 d x}{2}\right )-2835 \sin \left (5 c+\frac{7 d x}{2}\right )+4149 \sin \left (3 c+\frac{9 d x}{2}\right )+252 \sin \left (4 c+\frac{9 d x}{2}\right )+3582 \sin \left (5 c+\frac{9 d x}{2}\right )-315 \sin \left (6 c+\frac{9 d x}{2}\right )+496 \sin \left (4 c+\frac{11 d x}{2}\right )+63 \sin \left (5 c+\frac{11 d x}{2}\right )+433 \sin \left (6 c+\frac{11 d x}{2}\right )-33978 \sin \left (\frac{d x}{2}\right )+52002 \sin \left (\frac{3 d x}{2}\right )\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \sec (c+d x)}{2016 d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]

[Out]

(160*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) - (160*Cos
[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) + (Cos[c/2 + (d*x)
/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]*(-33978*Sin[(d*x)/2] + 52002*Sin[(3*d*x)/2] - 56952*Sin[c - (d*x)/2] + 43722*
Sin[c + (d*x)/2] - 47208*Sin[2*c + (d*x)/2] - 18144*Sin[c + (3*d*x)/2] + 41796*Sin[2*c + (3*d*x)/2] - 28350*Si
n[3*c + (3*d*x)/2] + 34578*Sin[c + (5*d*x)/2] - 5691*Sin[2*c + (5*d*x)/2] + 28719*Sin[3*c + (5*d*x)/2] - 11550
*Sin[4*c + (5*d*x)/2] + 15517*Sin[2*c + (7*d*x)/2] - 504*Sin[3*c + (7*d*x)/2] + 13186*Sin[4*c + (7*d*x)/2] - 2
835*Sin[5*c + (7*d*x)/2] + 4149*Sin[3*c + (9*d*x)/2] + 252*Sin[4*c + (9*d*x)/2] + 3582*Sin[5*c + (9*d*x)/2] -
315*Sin[6*c + (9*d*x)/2] + 496*Sin[4*c + (11*d*x)/2] + 63*Sin[5*c + (11*d*x)/2] + 433*Sin[6*c + (11*d*x)/2]))/
(2016*d*(a + a*Cos[c + d*x])^5)

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Maple [A]  time = 0.068, size = 177, normalized size = 1.1 \begin{align*}{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{1}{14\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{129}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+5\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{5}}}-{\frac{1}{d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-5\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+cos(d*x+c)*a)^5,x)

[Out]

1/144/d/a^5*tan(1/2*d*x+1/2*c)^9+1/14/d/a^5*tan(1/2*d*x+1/2*c)^7+3/8/d/a^5*tan(1/2*d*x+1/2*c)^5+3/2/d/a^5*tan(
1/2*d*x+1/2*c)^3+129/16/d/a^5*tan(1/2*d*x+1/2*c)-1/d/a^5/(tan(1/2*d*x+1/2*c)-1)+5/d/a^5*ln(tan(1/2*d*x+1/2*c)-
1)-1/d/a^5/(tan(1/2*d*x+1/2*c)+1)-5/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)

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Maxima [A]  time = 1.17826, size = 278, normalized size = 1.65 \begin{align*} \frac{\frac{2016 \, \sin \left (d x + c\right )}{{\left (a^{5} - \frac{a^{5} \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{8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac{5040 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac{5040 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{1008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

1/1008*(2016*sin(d*x + c)/((a^5 - a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (8127*sin(d*x
+ c)/(cos(d*x + c) + 1) + 1512*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 72*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 7*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 5040*log(sin(d*x + c)/
(cos(d*x + c) + 1) + 1)/a^5 + 5040*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^5)/d

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Fricas [A]  time = 1.76818, size = 755, normalized size = 4.49 \begin{align*} -\frac{315 \,{\left (\cos \left (d x + c\right )^{6} + 5 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (\cos \left (d x + c\right )^{6} + 5 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (496 \, \cos \left (d x + c\right )^{5} + 2165 \, \cos \left (d x + c\right )^{4} + 3633 \, \cos \left (d x + c\right )^{3} + 2840 \, \cos \left (d x + c\right )^{2} + 946 \, \cos \left (d x + c\right ) + 63\right )} \sin \left (d x + c\right )}{126 \,{\left (a^{5} d \cos \left (d x + c\right )^{6} + 5 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 5 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} d \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/126*(315*(cos(d*x + c)^6 + 5*cos(d*x + c)^5 + 10*cos(d*x + c)^4 + 10*cos(d*x + c)^3 + 5*cos(d*x + c)^2 + co
s(d*x + c))*log(sin(d*x + c) + 1) - 315*(cos(d*x + c)^6 + 5*cos(d*x + c)^5 + 10*cos(d*x + c)^4 + 10*cos(d*x +
c)^3 + 5*cos(d*x + c)^2 + cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(496*cos(d*x + c)^5 + 2165*cos(d*x + c)^4 +
3633*cos(d*x + c)^3 + 2840*cos(d*x + c)^2 + 946*cos(d*x + c) + 63)*sin(d*x + c))/(a^5*d*cos(d*x + c)^6 + 5*a^
5*d*cos(d*x + c)^5 + 10*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 5*a^5*d*cos(d*x + c)^2 + a^5*d*cos(d*
x + c))

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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(sec(d*x+c)**2/(a+a*cos(d*x+c))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.33544, size = 209, normalized size = 1.24 \begin{align*} -\frac{\frac{5040 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac{5040 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac{2016 \, \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^{5}} - \frac{7 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 72 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1512 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{1008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

-1/1008*(5040*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 - 5040*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5 + 2016*tan(
1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*a^5) - (7*a^40*tan(1/2*d*x + 1/2*c)^9 + 72*a^40*tan(1/2*d*x + 1
/2*c)^7 + 378*a^40*tan(1/2*d*x + 1/2*c)^5 + 1512*a^40*tan(1/2*d*x + 1/2*c)^3 + 8127*a^40*tan(1/2*d*x + 1/2*c))
/a^45)/d