\(\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [509]

Optimal result

Integrand size = 43, antiderivative size = 311 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1015 A+1132 B+1304 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \]

[Out]

1/512*a^(5/2)*(1015*A+1132*B+1304*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/60*a*(5*A+12*B)*cos
(d*x+c)^4*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/512*a^3
*(1015*A+1132*B+1304*C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/768*a^3*(1015*A+1132*B+1304*C)*cos(d*x+c)*sin(d*
x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/960*a^3*(545*A+628*B+680*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1
/480*a^2*(115*A+156*B+120*C)*cos(d*x+c)^3*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4171, 4102, 4100, 3890, 3859, 209} \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1015 A+1132 B+1304 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{512 d}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{960 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x) \cos (c+d x)}{768 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{480 d}+\frac {a (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]

[In]

Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1
015*A + 1132*B + 1304*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Cos[
c + d*x]*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Cos[c + d*x]^2*Sin[c +
d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*
Sin[c + d*x])/(480*d) + (a*(5*A + 12*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (A*Co
s[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3890

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

Rule 4100

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

Rule 4102

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

Rule 4171

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

Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (5 A+12 B)+\frac {1}{2} a (5 A+12 C) \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (115 A+156 B+120 C)+\frac {15}{4} a^2 (5 A+4 B+8 C) \sec (c+d x)\right ) \, dx}{30 a} \\ & = \frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{8} a^3 (545 A+628 B+680 C)+\frac {5}{8} a^3 (235 A+252 B+312 C) \sec (c+d x)\right ) \, dx}{120 a} \\ & = \frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {1}{384} \left (a^2 (1015 A+1132 B+1304 C)\right ) \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {1}{512} \left (a^2 (1015 A+1132 B+1304 C)\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\left (a^2 (1015 A+1132 B+1304 C)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx}{1024} \\ & = \frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}-\frac {\left (a^3 (1015 A+1132 B+1304 C)\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d} \\ & = \frac {a^{5/2} (1015 A+1132 B+1304 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 2.57 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.04 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (862155 A \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+933660 B \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+1020600 C \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+449183 A \sqrt {1-\sec (c+d x)}+419436 B \sqrt {1-\sec (c+d x)}+349272 C \sqrt {1-\sec (c+d x)}+1358777 A \cos (c+d x) \sqrt {1-\sec (c+d x)}+1333044 B \cos (c+d x) \sqrt {1-\sec (c+d x)}+1283688 C \cos (c+d x) \sqrt {1-\sec (c+d x)}+505449 A \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+455508 B \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+352296 C \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+190834 A \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+137448 B \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+87696 C \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+57666 A \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+36072 B \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+3024 C \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+15176 A \cos (5 (c+d x)) \sqrt {1-\sec (c+d x)}+2592 B \cos (5 (c+d x)) \sqrt {1-\sec (c+d x)}+1400 A \cos (6 (c+d x)) \sqrt {1-\sec (c+d x)}+774144 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+552960 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},6,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+430080 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},7,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{483840 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^2*(862155*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 933660*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 1020600*C*ArcTanh[
Sqrt[1 - Sec[c + d*x]]] + 449183*A*Sqrt[1 - Sec[c + d*x]] + 419436*B*Sqrt[1 - Sec[c + d*x]] + 349272*C*Sqrt[1
- Sec[c + d*x]] + 1358777*A*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 1333044*B*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]
] + 1283688*C*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 505449*A*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 455508*
B*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 352296*C*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 190834*A*Cos[3*
(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 137448*B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 87696*C*Cos[3*(c + d*x)
]*Sqrt[1 - Sec[c + d*x]] + 57666*A*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 36072*B*Cos[4*(c + d*x)]*Sqrt[1 -
Sec[c + d*x]] + 3024*C*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 15176*A*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*
x]] + 2592*B*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1400*A*Cos[6*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 774144
*C*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 552960*B*Hypergeometric2F1[1/2, 6
, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 430080*A*Hypergeometric2F1[1/2, 7, 3/2, 1 - Sec[c + d*x]]*Sq
rt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(483840*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d
*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(664\) vs. \(2(279)=558\).

Time = 0.57 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.14

\[\frac {a^{2} \left (1280 A \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )+4480 A \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )+1536 B \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )+6960 A \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+5568 B \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+1920 C \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+8120 A \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+9056 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}+7360 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+15225 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+10150 A \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+16980 B \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+11320 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}+19560 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+13040 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+15225 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+15225 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+16980 B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+16980 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+19560 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+19560 C \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{7680 d \left (\cos \left (d x +c \right )+1\right )}\]

[In]

int(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

1/7680*a^2/d*(1280*A*cos(d*x+c)^6*sin(d*x+c)+4480*A*cos(d*x+c)^5*sin(d*x+c)+1536*B*cos(d*x+c)^5*sin(d*x+c)+696
0*A*cos(d*x+c)^4*sin(d*x+c)+5568*B*cos(d*x+c)^4*sin(d*x+c)+1920*C*cos(d*x+c)^4*sin(d*x+c)+8120*A*cos(d*x+c)^3*
sin(d*x+c)+9056*B*sin(d*x+c)*cos(d*x+c)^3+7360*C*cos(d*x+c)^3*sin(d*x+c)+15225*A*(-cos(d*x+c)/(cos(d*x+c)+1))^
(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+10150*A*cos(d*x+c)^2*si
n(d*x+c)+16980*B*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*cos(d*x+c)+11320*B*sin(d*x+c)*cos(d*x+c)^2+19560*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(
d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+13040*C*cos(d*x+c)^2*sin(d*x+c)+15225*A*(
-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+15225*
A*cos(d*x+c)*sin(d*x+c)+16980*B*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x
+c)/(cos(d*x+c)+1))^(1/2))+16980*B*cos(d*x+c)*sin(d*x+c)+19560*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(si
n(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+19560*C*cos(d*x+c)*sin(d*x+c))*(a*(1+sec(d*x+c)))^
(1/2)/(cos(d*x+c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.72 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {15 \, {\left ({\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (1280 \, A a^{2} \cos \left (d x + c\right )^{6} + 128 \, {\left (35 \, A + 12 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (145 \, A + 116 \, B + 40 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (1015 \, A + 1132 \, B + 920 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15360 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (1280 \, A a^{2} \cos \left (d x + c\right )^{6} + 128 \, {\left (35 \, A + 12 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (145 \, A + 116 \, B + 40 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (1015 \, A + 1132 \, B + 920 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/15360*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 1132*B + 1304*C)*a^2)*sqrt(-a)*log((2*a*
cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c)
- a)/(cos(d*x + c) + 1)) + 2*(1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 12*B)*a^2*cos(d*x + c)^5 + 48*(145*A +
116*B + 40*C)*a^2*cos(d*x + c)^4 + 8*(1015*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 1304
*C)*a^2*cos(d*x + c)^2 + 15*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c
))*sin(d*x + c))/(d*cos(d*x + c) + d), -1/7680*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 11
32*B + 1304*C)*a^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))
) - (1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 12*B)*a^2*cos(d*x + c)^5 + 48*(145*A + 116*B + 40*C)*a^2*cos(d*x
+ c)^4 + 8*(1015*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)^2 + 1
5*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*
x + c) + d)]

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]

[In]

int(cos(c + d*x)^6*(a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

int(cos(c + d*x)^6*(a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)