3.146 \(\int \frac{\sec ^5(c+d x) (A+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=232 \[ -\frac{32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac{(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(10 A+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{16 (5 A+54 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(2 A+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(A+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]

[Out]

((2*A + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(5*A + 54*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Se
c[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x]
)^2) - (16*(5*A + 54*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^5*
Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)

________________________________________________________________________________________

Rubi [A]  time = 0.64895, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4085, 4019, 3787, 3767, 8, 3768, 3770} \[ -\frac{32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac{(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(10 A+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{16 (5 A+54 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(2 A+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(A+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

((2*A + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(5*A + 54*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Se
c[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x]
)^2) - (16*(5*A + 54*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^5*
Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)

Rule 4085

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> -Simp[(a*(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(a*f*(
2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*C*n + A*b*(
2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x]
&& EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

Rule 4019

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

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, 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 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 ^5(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{\int \frac{\sec ^5(c+d x) (-a (2 A-5 C)-a (2 A+9 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^4(c+d x) \left (56 a^2 C-a^2 (10 A+73 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x) \left (3 a^3 (10 A+129 C)-a^3 (50 A+477 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac{\int \sec ^2(c+d x) \left (32 a^4 (5 A+54 C)-105 a^4 (2 A+21 C) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac{(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{(2 A+21 C) \int \sec ^3(c+d x) \, dx}{a^4}-\frac{(32 (5 A+54 C)) \int \sec ^2(c+d x) \, dx}{105 a^4}\\ &=\frac{(2 A+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{(2 A+21 C) \int \sec (c+d x) \, dx}{2 a^4}+\frac{(32 (5 A+54 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=\frac{(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac{(2 A+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac{16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 4.22883, size = 746, normalized size = 3.22 \[ -\frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\sec \left (\frac{c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (-17220 A \sin \left (c-\frac{d x}{2}\right )+17220 A \sin \left (c+\frac{d x}{2}\right )-14140 A \sin \left (2 c+\frac{d x}{2}\right )-9800 A \sin \left (c+\frac{3 d x}{2}\right )+15160 A \sin \left (2 c+\frac{3 d x}{2}\right )-9800 A \sin \left (3 c+\frac{3 d x}{2}\right )+10920 A \sin \left (c+\frac{5 d x}{2}\right )-4760 A \sin \left (2 c+\frac{5 d x}{2}\right )+10920 A \sin \left (3 c+\frac{5 d x}{2}\right )-4760 A \sin \left (4 c+\frac{5 d x}{2}\right )+5890 A \sin \left (2 c+\frac{7 d x}{2}\right )-1470 A \sin \left (3 c+\frac{7 d x}{2}\right )+5890 A \sin \left (4 c+\frac{7 d x}{2}\right )-1470 A \sin \left (5 c+\frac{7 d x}{2}\right )+2030 A \sin \left (3 c+\frac{9 d x}{2}\right )-210 A \sin \left (4 c+\frac{9 d x}{2}\right )+2030 A \sin \left (5 c+\frac{9 d x}{2}\right )-210 A \sin \left (6 c+\frac{9 d x}{2}\right )+320 A \sin \left (4 c+\frac{11 d x}{2}\right )+320 A \sin \left (6 c+\frac{11 d x}{2}\right )-14 (1010 A+5229 C) \sin \left (\frac{d x}{2}\right )+4 (3790 A+41667 C) \sin \left (\frac{3 d x}{2}\right )-183162 C \sin \left (c-\frac{d x}{2}\right )+100842 C \sin \left (c+\frac{d x}{2}\right )-155526 C \sin \left (2 c+\frac{d x}{2}\right )-37380 C \sin \left (c+\frac{3 d x}{2}\right )+101148 C \sin \left (2 c+\frac{3 d x}{2}\right )-102900 C \sin \left (3 c+\frac{3 d x}{2}\right )+119364 C \sin \left (c+\frac{5 d x}{2}\right )-8820 C \sin \left (2 c+\frac{5 d x}{2}\right )+78204 C \sin \left (3 c+\frac{5 d x}{2}\right )-49980 C \sin \left (4 c+\frac{5 d x}{2}\right )+64053 C \sin \left (2 c+\frac{7 d x}{2}\right )+3885 C \sin \left (3 c+\frac{7 d x}{2}\right )+44733 C \sin \left (4 c+\frac{7 d x}{2}\right )-15435 C \sin \left (5 c+\frac{7 d x}{2}\right )+21987 C \sin \left (3 c+\frac{9 d x}{2}\right )+3675 C \sin \left (4 c+\frac{9 d x}{2}\right )+16107 C \sin \left (5 c+\frac{9 d x}{2}\right )-2205 C \sin \left (6 c+\frac{9 d x}{2}\right )+3456 C \sin \left (4 c+\frac{11 d x}{2}\right )+840 C \sin \left (5 c+\frac{11 d x}{2}\right )+2616 C \sin \left (6 c+\frac{11 d x}{2}\right )\right )+53760 (2 A+21 C) \cos ^7\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{3360 a^4 d (\sec (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

-(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(53760*(2*A + 21*C)*Cos[(c + d*x)/2]^7*(Log[Cos[(c +
d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-14*
(1010*A + 5229*C)*Sin[(d*x)/2] + 4*(3790*A + 41667*C)*Sin[(3*d*x)/2] - 17220*A*Sin[c - (d*x)/2] - 183162*C*Sin
[c - (d*x)/2] + 17220*A*Sin[c + (d*x)/2] + 100842*C*Sin[c + (d*x)/2] - 14140*A*Sin[2*c + (d*x)/2] - 155526*C*S
in[2*c + (d*x)/2] - 9800*A*Sin[c + (3*d*x)/2] - 37380*C*Sin[c + (3*d*x)/2] + 15160*A*Sin[2*c + (3*d*x)/2] + 10
1148*C*Sin[2*c + (3*d*x)/2] - 9800*A*Sin[3*c + (3*d*x)/2] - 102900*C*Sin[3*c + (3*d*x)/2] + 10920*A*Sin[c + (5
*d*x)/2] + 119364*C*Sin[c + (5*d*x)/2] - 4760*A*Sin[2*c + (5*d*x)/2] - 8820*C*Sin[2*c + (5*d*x)/2] + 10920*A*S
in[3*c + (5*d*x)/2] + 78204*C*Sin[3*c + (5*d*x)/2] - 4760*A*Sin[4*c + (5*d*x)/2] - 49980*C*Sin[4*c + (5*d*x)/2
] + 5890*A*Sin[2*c + (7*d*x)/2] + 64053*C*Sin[2*c + (7*d*x)/2] - 1470*A*Sin[3*c + (7*d*x)/2] + 3885*C*Sin[3*c
+ (7*d*x)/2] + 5890*A*Sin[4*c + (7*d*x)/2] + 44733*C*Sin[4*c + (7*d*x)/2] - 1470*A*Sin[5*c + (7*d*x)/2] - 1543
5*C*Sin[5*c + (7*d*x)/2] + 2030*A*Sin[3*c + (9*d*x)/2] + 21987*C*Sin[3*c + (9*d*x)/2] - 210*A*Sin[4*c + (9*d*x
)/2] + 3675*C*Sin[4*c + (9*d*x)/2] + 2030*A*Sin[5*c + (9*d*x)/2] + 16107*C*Sin[5*c + (9*d*x)/2] - 210*A*Sin[6*
c + (9*d*x)/2] - 2205*C*Sin[6*c + (9*d*x)/2] + 320*A*Sin[4*c + (11*d*x)/2] + 3456*C*Sin[4*c + (11*d*x)/2] + 84
0*C*Sin[5*c + (11*d*x)/2] + 320*A*Sin[6*c + (11*d*x)/2] + 2616*C*Sin[6*c + (11*d*x)/2])))/(3360*a^4*d*(A + 2*C
+ A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^4)

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Maple [A]  time = 0.08, size = 329, normalized size = 1.4 \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{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{9\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{11\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{13\,C}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{111\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{A}{d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{21\,C}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{C}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{9\,C}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{A}{d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{21\,C}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{C}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{9\,C}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5*(A+C*sec(d*x+c)^2)/(a+a*sec(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*C*tan(1/2*d*x+1/2*c)^7-1/8/d/a^4*tan(1/2*d*x+1/2*c)^5*A-9/40/d/a
^4*C*tan(1/2*d*x+1/2*c)^5-11/24/d/a^4*A*tan(1/2*d*x+1/2*c)^3-13/8/d/a^4*C*tan(1/2*d*x+1/2*c)^3-15/8/d/a^4*A*ta
n(1/2*d*x+1/2*c)-111/8/d/a^4*C*tan(1/2*d*x+1/2*c)+1/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*A+21/2/d/a^4*ln(tan(1/2*d*x
+1/2*c)+1)*C-1/2/d/a^4*C/(tan(1/2*d*x+1/2*c)+1)^2+9/2/d/a^4*C/(tan(1/2*d*x+1/2*c)+1)-1/d/a^4*ln(tan(1/2*d*x+1/
2*c)-1)*A-21/2/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*C+1/2/d/a^4*C/(tan(1/2*d*x+1/2*c)-1)^2+9/2/d/a^4*C/(tan(1/2*d*x+
1/2*c)-1)

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Maxima [A]  time = 0.97384, size = 502, normalized size = 2.16 \begin{align*} -\frac{3 \, C{\left (\frac{280 \,{\left (\frac{7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac{2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \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}}}{a^{4}} - \frac{2940 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{2940 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + 5 \, A{\left (\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{77 \, \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{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{168 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{168 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/840*(3*C*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 - 2*a^4*sin(
d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c)
+ 1) + 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(co
s(d*x + c) + 1)^7)/a^4 - 2940*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 2940*log(sin(d*x + c)/(cos(d*x +
c) + 1) - 1)/a^4) + 5*A*((315*sin(d*x + c)/(cos(d*x + c) + 1) + 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*si
n(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 168*log(sin(d*x + c)/(cos(d*x
+ c) + 1) + 1)/a^4 + 168*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d

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Fricas [A]  time = 0.53375, size = 923, normalized size = 3.98 \begin{align*} \frac{105 \,{\left ({\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left ({\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (64 \,{\left (5 \, A + 54 \, C\right )} \cos \left (d x + c\right )^{5} +{\left (1070 \, A + 11619 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (310 \, A + 3411 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (130 \, A + 1509 \, C\right )} \cos \left (d x + c\right )^{2} + 420 \, C \cos \left (d x + c\right ) - 105 \, C\right )} \sin \left (d x + c\right )}{420 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/420*(105*((2*A + 21*C)*cos(d*x + c)^6 + 4*(2*A + 21*C)*cos(d*x + c)^5 + 6*(2*A + 21*C)*cos(d*x + c)^4 + 4*(2
*A + 21*C)*cos(d*x + c)^3 + (2*A + 21*C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 105*((2*A + 21*C)*cos(d*x + c
)^6 + 4*(2*A + 21*C)*cos(d*x + c)^5 + 6*(2*A + 21*C)*cos(d*x + c)^4 + 4*(2*A + 21*C)*cos(d*x + c)^3 + (2*A + 2
1*C)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(64*(5*A + 54*C)*cos(d*x + c)^5 + (1070*A + 11619*C)*cos(d*x +
c)^4 + 4*(310*A + 3411*C)*cos(d*x + c)^3 + 4*(130*A + 1509*C)*cos(d*x + c)^2 + 420*C*cos(d*x + c) - 105*C)*si
n(d*x + c))/(a^4*d*cos(d*x + c)^6 + 4*a^4*d*cos(d*x + c)^5 + 6*a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 +
a^4*d*cos(d*x + c)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{A \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec{\left (c + d x \right )} + 1}\, dx}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**5*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**4,x)

[Out]

(Integral(A*sec(c + d*x)**5/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x)
+ Integral(C*sec(c + d*x)**7/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1),
x))/a**4

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Giac [A]  time = 1.23292, size = 325, normalized size = 1.4 \begin{align*} \frac{\frac{420 \,{\left (2 \, A + 21 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{420 \,{\left (2 \, A + 21 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{840 \,{\left (9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, C \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^{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} + 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 189 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 11655 \, 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(sec(d*x+c)^5*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/840*(420*(2*A + 21*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 420*(2*A + 21*C)*log(abs(tan(1/2*d*x + 1/2*c)
- 1))/a^4 + 840*(9*C*tan(1/2*d*x + 1/2*c)^3 - 7*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*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 + 105*A*a^24*tan(1/2*d*x + 1/2*c)^5 + 1
89*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 385*A*a^24*tan(1/2*d*x + 1/2*c)^3 + 1365*C*a^24*tan(1/2*d*x + 1/2*c)^3 + 15
75*A*a^24*tan(1/2*d*x + 1/2*c) + 11655*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d