3.92 \(\int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=224 \[ -\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac {31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac {3832 \tan (c+d x) \sec (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

[Out]

31/2*arctanh(sin(d*x+c))/a^5/d-7664/315*tan(d*x+c)/a^5/d+31/2*sec(d*x+c)*tan(d*x+c)/a^5/d-1/9*sec(d*x+c)*tan(d
*x+c)/d/(a+a*cos(d*x+c))^5-17/63*sec(d*x+c)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^4-28/45*sec(d*x+c)*tan(d*x+c)/a^2/
d/(a+a*cos(d*x+c))^3-577/315*sec(d*x+c)*tan(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-3832/315*sec(d*x+c)*tan(d*x+c)/d/(
a^5+a^5*cos(d*x+c))

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Rubi [A]  time = 0.54, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ -\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac {31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac {3832 \tan (c+d x) \sec (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]

[Out]

(31*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (7664*Tan[c + d*x])/(315*a^5*d) + (31*Sec[c + d*x]*Tan[c + d*x])/(2*a^5
*d) - (Sec[c + d*x]*Tan[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Sec[c + d*x]*Tan[c + d*x])/(63*a*d*(a + a
*Cos[c + d*x])^4) - (28*Sec[c + d*x]*Tan[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Sec[c + d*x]*Tan[c
+ d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Sec[c + d*x]*Tan[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x])
)

Rule 8

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

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 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 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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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 ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {\int \frac {(11 a-6 a \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\int \frac {\left (111 a^2-85 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (947 a^3-784 a^3 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (6303 a^4-5193 a^4 \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {\int \left (29295 a^5-22992 a^5 \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{945 a^{10}}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac {7664 \int \sec ^2(c+d x) \, dx}{315 a^5}+\frac {31 \int \sec ^3(c+d x) \, dx}{a^5}\\ &=\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {31 \int \sec (c+d x) \, dx}{2 a^5}+\frac {7664 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{315 a^5 d}\\ &=\frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}

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Mathematica [B]  time = 6.35, size = 507, normalized size = 2.26 \[ -\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}+\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}+\frac {\sec \left (\frac {c}{2}\right ) \sec (c) \left (3057654 \sin \left (c-\frac {d x}{2}\right )-1885854 \sin \left (c+\frac {d x}{2}\right )+2644362 \sin \left (2 c+\frac {d x}{2}\right )+867048 \sin \left (c+\frac {3 d x}{2}\right )-1868436 \sin \left (2 c+\frac {3 d x}{2}\right )+1821498 \sin \left (3 c+\frac {3 d x}{2}\right )-2083537 \sin \left (c+\frac {5 d x}{2}\right )+339885 \sin \left (2 c+\frac {5 d x}{2}\right )-1456687 \sin \left (3 c+\frac {5 d x}{2}\right )+966735 \sin \left (4 c+\frac {5 d x}{2}\right )-1195641 \sin \left (2 c+\frac {7 d x}{2}\right )+46515 \sin \left (3 c+\frac {7 d x}{2}\right )-874341 \sin \left (4 c+\frac {7 d x}{2}\right )+367815 \sin \left (5 c+\frac {7 d x}{2}\right )-494579 \sin \left (3 c+\frac {9 d x}{2}\right )-31815 \sin \left (4 c+\frac {9 d x}{2}\right )-374879 \sin \left (5 c+\frac {9 d x}{2}\right )+87885 \sin \left (6 c+\frac {9 d x}{2}\right )-128187 \sin \left (4 c+\frac {11 d x}{2}\right )-18585 \sin \left (5 c+\frac {11 d x}{2}\right )-99837 \sin \left (6 c+\frac {11 d x}{2}\right )+9765 \sin \left (7 c+\frac {11 d x}{2}\right )-15328 \sin \left (5 c+\frac {13 d x}{2}\right )-3150 \sin \left (6 c+\frac {13 d x}{2}\right )-12178 \sin \left (7 c+\frac {13 d x}{2}\right )+1472562 \sin \left (\frac {d x}{2}\right )-2822886 \sin \left (\frac {3 d x}{2}\right )\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{40320 d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]

[Out]

(-496*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) + (496*Co
s[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]^2*(1472562*Sin[(d*x)/2] - 2822886*Sin[(3*d*x)/2] + 3057654*Sin[c - (d*x)/2]
- 1885854*Sin[c + (d*x)/2] + 2644362*Sin[2*c + (d*x)/2] + 867048*Sin[c + (3*d*x)/2] - 1868436*Sin[2*c + (3*d*x
)/2] + 1821498*Sin[3*c + (3*d*x)/2] - 2083537*Sin[c + (5*d*x)/2] + 339885*Sin[2*c + (5*d*x)/2] - 1456687*Sin[3
*c + (5*d*x)/2] + 966735*Sin[4*c + (5*d*x)/2] - 1195641*Sin[2*c + (7*d*x)/2] + 46515*Sin[3*c + (7*d*x)/2] - 87
4341*Sin[4*c + (7*d*x)/2] + 367815*Sin[5*c + (7*d*x)/2] - 494579*Sin[3*c + (9*d*x)/2] - 31815*Sin[4*c + (9*d*x
)/2] - 374879*Sin[5*c + (9*d*x)/2] + 87885*Sin[6*c + (9*d*x)/2] - 128187*Sin[4*c + (11*d*x)/2] - 18585*Sin[5*c
+ (11*d*x)/2] - 99837*Sin[6*c + (11*d*x)/2] + 9765*Sin[7*c + (11*d*x)/2] - 15328*Sin[5*c + (13*d*x)/2] - 3150
*Sin[6*c + (13*d*x)/2] - 12178*Sin[7*c + (13*d*x)/2]))/(40320*d*(a + a*Cos[c + d*x])^5)

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fricas [A]  time = 1.14, size = 294, normalized size = 1.31 \[ \frac {9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15328 \, \cos \left (d x + c\right )^{6} + 66875 \, \cos \left (d x + c\right )^{5} + 112119 \, \cos \left (d x + c\right )^{4} + 87440 \, \cos \left (d x + c\right )^{3} + 28828 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right ) - 315\right )} \sin \left (d x + c\right )}{1260 \, {\left (a^{5} d \cos \left (d x + c\right )^{7} + 5 \, a^{5} d \cos \left (d x + c\right )^{6} + 10 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 5 \, a^{5} d \cos \left (d x + c\right )^{3} + a^{5} d \cos \left (d x + c\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/1260*(9765*(cos(d*x + c)^7 + 5*cos(d*x + c)^6 + 10*cos(d*x + c)^5 + 10*cos(d*x + c)^4 + 5*cos(d*x + c)^3 + c
os(d*x + c)^2)*log(sin(d*x + c) + 1) - 9765*(cos(d*x + c)^7 + 5*cos(d*x + c)^6 + 10*cos(d*x + c)^5 + 10*cos(d*
x + c)^4 + 5*cos(d*x + c)^3 + cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(15328*cos(d*x + c)^6 + 66875*cos(d*x
+ c)^5 + 112119*cos(d*x + c)^4 + 87440*cos(d*x + c)^3 + 28828*cos(d*x + c)^2 + 1575*cos(d*x + c) - 315)*sin(d
*x + c))/(a^5*d*cos(d*x + c)^7 + 5*a^5*d*cos(d*x + c)^6 + 10*a^5*d*cos(d*x + c)^5 + 10*a^5*d*cos(d*x + c)^4 +
5*a^5*d*cos(d*x + c)^3 + a^5*d*cos(d*x + c)^2)

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giac [A]  time = 0.79, size = 171, normalized size = 0.76 \[ \frac {\frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(78120*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 - 78120*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5 + 5040*(11
*tan(1/2*d*x + 1/2*c)^3 - 9*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^5) - (35*a^40*tan(1/2*d*x
+ 1/2*c)^9 + 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1/2*c)^5 + 15750*a^40*tan(1/2*d*x + 1/2
*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

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maple [A]  time = 0.11, size = 219, normalized size = 0.98 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{5}}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{5}}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{5}}-\frac {351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}+\frac {1}{2 d \,a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {11}{2 d \,a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{5}}-\frac {1}{2 d \,a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{2 d \,a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(a+a*cos(d*x+c))^5,x)

[Out]

-1/144/d/a^5*tan(1/2*d*x+1/2*c)^9-5/56/d/a^5*tan(1/2*d*x+1/2*c)^7-3/5/d/a^5*tan(1/2*d*x+1/2*c)^5-25/8/d/a^5*ta
n(1/2*d*x+1/2*c)^3-351/16/d/a^5*tan(1/2*d*x+1/2*c)+1/2/d/a^5/(tan(1/2*d*x+1/2*c)-1)^2+11/2/d/a^5/(tan(1/2*d*x+
1/2*c)-1)-31/2/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)-1/2/d/a^5/(tan(1/2*d*x+1/2*c)+1)^2+11/2/d/a^5/(tan(1/2*d*x+1/2*c
)+1)+31/2/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)

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maxima [A]  time = 0.69, size = 251, normalized size = 1.12 \[ -\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} - \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{5040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) - 11*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^5 - 2*a^5*sin(d*
x + c)^2/(cos(d*x + c) + 1)^2 + a^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c)
+ 1) + 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 450*sin(d*x + c)
^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 78120*log(sin(d*x + c)/(cos(d*x + c) +
1) + 1)/a^5 + 78120*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^5)/d

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mupad [B]  time = 0.48, size = 179, normalized size = 0.80 \[ \frac {31\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^5\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,a^5\,d}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^5\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{144\,a^5\,d}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^5\,d}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^5\right )}-\frac {351\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^5\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^3*(a + a*cos(c + d*x))^5),x)

[Out]

(31*atanh(tan(c/2 + (d*x)/2)))/(a^5*d) - (3*tan(c/2 + (d*x)/2)^5)/(5*a^5*d) - (5*tan(c/2 + (d*x)/2)^7)/(56*a^5
*d) - tan(c/2 + (d*x)/2)^9/(144*a^5*d) - (25*tan(c/2 + (d*x)/2)^3)/(8*a^5*d) - (9*tan(c/2 + (d*x)/2) - 11*tan(
c/2 + (d*x)/2)^3)/(d*(a^5*tan(c/2 + (d*x)/2)^4 - 2*a^5*tan(c/2 + (d*x)/2)^2 + a^5)) - (351*tan(c/2 + (d*x)/2))
/(16*a^5*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\cos ^{5}{\left (c + d x \right )} + 5 \cos ^{4}{\left (c + d x \right )} + 10 \cos ^{3}{\left (c + d x \right )} + 10 \cos ^{2}{\left (c + d x \right )} + 5 \cos {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3/(a+a*cos(d*x+c))**5,x)

[Out]

Integral(sec(c + d*x)**3/(cos(c + d*x)**5 + 5*cos(c + d*x)**4 + 10*cos(c + d*x)**3 + 10*cos(c + d*x)**2 + 5*co
s(c + d*x) + 1), x)/a**5

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