\(\int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [108]

Optimal result

Integrand size = 31, antiderivative size = 238 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]

[Out]

-1/2*(8*A-21*B)*arctanh(sin(d*x+c))/a^4/d+8/105*(83*A-216*B)*tan(d*x+c)/a^4/d-1/2*(8*A-21*B)*sec(d*x+c)*tan(d*
x+c)/a^4/d+1/105*(52*A-129*B)*sec(d*x+c)^3*tan(d*x+c)/a^4/d/(1+sec(d*x+c))^2+4/105*(83*A-216*B)*sec(d*x+c)^2*t
an(d*x+c)/a^4/d/(1+sec(d*x+c))+1/7*(A-B)*sec(d*x+c)^5*tan(d*x+c)/d/(a+a*sec(d*x+c))^4+1/5*(A-2*B)*sec(d*x+c)^4
*tan(d*x+c)/a/d/(a+a*sec(d*x+c))^3

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4104, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}+\frac {(52 A-129 B) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {4 (83 A-216 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(8 A-21 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac {(A-B) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(A-2 B) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]

[In]

Int[(Sec[c + d*x]^6*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

-1/2*((8*A - 21*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) + (8*(83*A - 216*B)*Tan[c + d*x])/(105*a^4*d) - ((8*A - 21*B
)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) + ((52*A - 129*B)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c +
d*x])^2) + (4*(83*A - 216*B)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*
x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c
+ d*x])^3)

Rule 8

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

Rule 3852

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 3853

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 3855

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

Rule 3872

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 4104

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \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) (5 a (A-B)-a (2 A-9 B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \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 (28 a^2 (A-2 B)-a^2 (24 A-73 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \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 (52 A-129 B)-a^3 (176 A-477 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \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 (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(8 (83 A-216 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac {(8 A-21 B) \int \sec ^3(c+d x) \, dx}{a^4} \\ & = -\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(8 A-21 B) \int \sec (c+d x) \, dx}{2 a^4}-\frac {(8 (83 A-216 B)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = -\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec ^4(c+d x) \left (-13440 (8 A-21 B) \text {arctanh}(\sin (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right )+(22888 A-58161 B+8 (4994 A-12813 B) \cos (c+d x)+60 (456 A-1177 B) \cos (2 (c+d x))+13864 A \cos (3 (c+d x))-35928 B \cos (3 (c+d x))+4472 A \cos (4 (c+d x))-11619 B \cos (4 (c+d x))+664 A \cos (5 (c+d x))-1728 B \cos (5 (c+d x))) \sec (c+d x) \tan (c+d x)\right )}{1680 a^4 d (1+\sec (c+d x))^4} \]

[In]

Integrate[(Sec[c + d*x]^6*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

(Sec[c + d*x]^4*(-13440*(8*A - 21*B)*ArcTanh[Sin[c + d*x]]*Cos[(c + d*x)/2]^8 + (22888*A - 58161*B + 8*(4994*A
- 12813*B)*Cos[c + d*x] + 60*(456*A - 1177*B)*Cos[2*(c + d*x)] + 13864*A*Cos[3*(c + d*x)] - 35928*B*Cos[3*(c
+ d*x)] + 4472*A*Cos[4*(c + d*x)] - 11619*B*Cos[4*(c + d*x)] + 664*A*Cos[5*(c + d*x)] - 1728*B*Cos[5*(c + d*x)
])*Sec[c + d*x]*Tan[c + d*x]))/(1680*a^4*d*(1 + Sec[c + d*x])^4)

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\frac {3360 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {21 B}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3360 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {21 B}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+559 \left (\frac {15 \left (228 A -\frac {1177 B}{2}\right ) \cos \left (2 d x +2 c \right )}{559}+\frac {\left (1733 A -4491 B \right ) \cos \left (3 d x +3 c \right )}{559}+\left (A -\frac {11619 B}{4472}\right ) \cos \left (4 d x +4 c \right )+\frac {\left (83 A -216 B \right ) \cos \left (5 d x +5 c \right )}{559}+\frac {\left (4994 A -12813 B \right ) \cos \left (d x +c \right )}{559}+\frac {2861 A}{559}-\frac {58161 B}{4472}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{840 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(196\)
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-84 B +32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 B -32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\)	\(234\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-84 B +32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 B -32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\)	\(234\)
risch	\(\frac {i \left (840 A \,{\mathrm e}^{10 i \left (d x +c \right )}-2205 \,{\mathrm e}^{10 i \left (d x +c \right )} B +5880 A \,{\mathrm e}^{9 i \left (d x +c \right )}-15435 B \,{\mathrm e}^{9 i \left (d x +c \right )}+19040 A \,{\mathrm e}^{8 i \left (d x +c \right )}-49980 B \,{\mathrm e}^{8 i \left (d x +c \right )}+39200 A \,{\mathrm e}^{7 i \left (d x +c \right )}-102900 B \,{\mathrm e}^{7 i \left (d x +c \right )}+59248 A \,{\mathrm e}^{6 i \left (d x +c \right )}-155526 B \,{\mathrm e}^{6 i \left (d x +c \right )}+70896 A \,{\mathrm e}^{5 i \left (d x +c \right )}-183162 B \,{\mathrm e}^{5 i \left (d x +c \right )}+64384 A \,{\mathrm e}^{4 i \left (d x +c \right )}-166668 B \,{\mathrm e}^{4 i \left (d x +c \right )}+46032 A \,{\mathrm e}^{3 i \left (d x +c \right )}-119364 B \,{\mathrm e}^{3 i \left (d x +c \right )}+24664 A \,{\mathrm e}^{2 i \left (d x +c \right )}-64053 B \,{\mathrm e}^{2 i \left (d x +c \right )}+8456 \,{\mathrm e}^{i \left (d x +c \right )} A -21987 B \,{\mathrm e}^{i \left (d x +c \right )}+1328 A -3456 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{4} d}-\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{4} d}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a^{4} d}\)	\(372\)

[In]

int(sec(d*x+c)^6*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/840*(3360*(1+cos(2*d*x+2*c))*(A-21/8*B)*ln(tan(1/2*d*x+1/2*c)-1)-3360*(1+cos(2*d*x+2*c))*(A-21/8*B)*ln(tan(1
/2*d*x+1/2*c)+1)+559*(15/559*(228*A-1177/2*B)*cos(2*d*x+2*c)+1/559*(1733*A-4491*B)*cos(3*d*x+3*c)+(A-11619/447
2*B)*cos(4*d*x+4*c)+1/559*(83*A-216*B)*cos(5*d*x+5*c)+1/559*(4994*A-12813*B)*cos(d*x+c)+2861/559*A-58161/4472*
B)*sec(1/2*d*x+1/2*c)^6*tan(1/2*d*x+1/2*c))/d/a^4/(1+cos(2*d*x+2*c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.50 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (83 \, A - 216 \, B\right )} \cos \left (d x + c\right )^{5} + {\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 105 \, B\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(sec(d*x+c)^6*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/420*(105*((8*A - 21*B)*cos(d*x + c)^6 + 4*(8*A - 21*B)*cos(d*x + c)^5 + 6*(8*A - 21*B)*cos(d*x + c)^4 + 4*(
8*A - 21*B)*cos(d*x + c)^3 + (8*A - 21*B)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 105*((8*A - 21*B)*cos(d*x +
c)^6 + 4*(8*A - 21*B)*cos(d*x + c)^5 + 6*(8*A - 21*B)*cos(d*x + c)^4 + 4*(8*A - 21*B)*cos(d*x + c)^3 + (8*A -
21*B)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(16*(83*A - 216*B)*cos(d*x + c)^5 + (4472*A - 11619*B)*cos(d*
x + c)^4 + 4*(1318*A - 3411*B)*cos(d*x + c)^3 + 4*(592*A - 1509*B)*cos(d*x + c)^2 + 210*(A - 2*B)*cos(d*x + c)
+ 105*B)*sin(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)

Sympy [F]

\[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[In]

integrate(sec(d*x+c)**6*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**4,x)

[Out]

(Integral(A*sec(c + d*x)**6/(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(B*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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.76 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, B {\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 )} - A {\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 {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]

[In]

integrate(sec(d*x+c)^6*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/840*(3*B*(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) - A*(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 - 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^
4 + 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.12 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

integrate(sec(d*x+c)^6*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

-1/840*(420*(8*A - 21*B)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 420*(8*A - 21*B)*log(abs(tan(1/2*d*x + 1/2*c
) - 1))/a^4 + 840*(2*A*tan(1/2*d*x + 1/2*c)^3 - 9*B*tan(1/2*d*x + 1/2*c)^3 - 2*A*tan(1/2*d*x + 1/2*c) + 7*B*ta
n(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*B*a^24*tan(1
/2*d*x + 1/2*c)^7 + 147*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 189*B*a^24*tan(1/2*d*x + 1/2*c)^5 + 805*A*a^24*tan(1/2
*d*x + 1/2*c)^3 - 1365*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 5145*A*a^24*tan(1/2*d*x + 1/2*c) - 11655*B*a^24*tan(1/2
*d*x + 1/2*c))/a^28)/d

Mupad [B] (verification not implemented)

Time = 13.98 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.14 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {4\,A-6\,B}{8\,a^4}+\frac {5\,A-15\,B}{24\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{4\,a^4}-\frac {5\,B}{2\,a^4}+\frac {3\,\left (4\,A-6\,B\right )}{4\,a^4}+\frac {3\,\left (5\,A-15\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A-9\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A-7\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {4\,A-6\,B}{40\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,A-21\,B\right )}{a^4\,d} \]

[In]

int((A + B/cos(c + d*x))/(cos(c + d*x)^6*(a + a/cos(c + d*x))^4),x)

[Out]

(tan(c/2 + (d*x)/2)^3*((A - B)/(4*a^4) + (4*A - 6*B)/(8*a^4) + (5*A - 15*B)/(24*a^4)))/d + (tan(c/2 + (d*x)/2)
*((5*(A - B))/(4*a^4) - (5*B)/(2*a^4) + (3*(4*A - 6*B))/(4*a^4) + (3*(5*A - 15*B))/(8*a^4)))/d - (tan(c/2 + (d
*x)/2)^3*(2*A - 9*B) - tan(c/2 + (d*x)/2)*(2*A - 7*B))/(d*(a^4*tan(c/2 + (d*x)/2)^4 - 2*a^4*tan(c/2 + (d*x)/2)
^2 + a^4)) + (tan(c/2 + (d*x)/2)^5*((3*(A - B))/(40*a^4) + (4*A - 6*B)/(40*a^4)))/d + (tan(c/2 + (d*x)/2)^7*(A
- B))/(56*a^4*d) - (atanh(tan(c/2 + (d*x)/2))*(8*A - 21*B))/(a^4*d)