\(\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\) [390]
Optimal result
Integrand size = 43, antiderivative size = 263 \[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^{3/2} (133 A+150 B+176 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (133 A+150 B+176 C) \sec (c+d x) \tan (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}
\]
[Out]
1/128*a^(3/2)*(133*A+150*B+176*C)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/d+1/5*A*(a+a*cos(d*x+c))^
(3/2)*sec(d*x+c)^4*tan(d*x+c)/d+1/128*a^2*(133*A+150*B+176*C)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/192*a^2*(1
33*A+150*B+176*C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/240*a^2*(67*A+90*B+80*C)*sec(d*x+c)^2*tan(d
*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/40*a*(3*A+10*B)*sec(d*x+c)^3*(a+a*cos(d*x+c))^(1/2)*tan(d*x+c)/d
Rubi [A] (verified)
Time = 0.87 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3122, 3054, 3059, 2851, 2852,
212} \[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^{3/2} (133 A+150 B+176 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt {a \cos (c+d x)+a}}+\frac {a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{40 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}
\]
[In]
Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]
[Out]
(a^(3/2)*(133*A + 150*B + 176*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(133
*A + 150*B + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Sec[c + d*x]
*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*
d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (
A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2851
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x]
+ Dist[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Rule 2852
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[-2*(
b/f), Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 3054
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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a
*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*
n] || EqQ[c, 0])
Rule 3059
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Rule 3122
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(b*d*(n +
1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C
- B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x]
, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Rubi steps \begin{align*}
\text {integral}& = \frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (3 A+10 B)+\frac {5}{2} a (A+2 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{5 a} \\ & = \frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (67 A+90 B+80 C)+\frac {5}{4} a^2 (11 A+10 B+16 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{20 a} \\ & = \frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{96} (a (133 A+150 B+176 C)) \int \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \, dx \\ & = \frac {a^2 (133 A+150 B+176 C) \sec (c+d x) \tan (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{128} (a (133 A+150 B+176 C)) \int \sqrt {a+a \cos (c+d x)} \sec ^2(c+d x) \, dx \\ & = \frac {a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (133 A+150 B+176 C) \sec (c+d x) \tan (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{256} (a (133 A+150 B+176 C)) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (133 A+150 B+176 C) \sec (c+d x) \tan (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {\left (a^2 (133 A+150 B+176 C)\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d} \\ & = \frac {a^{3/2} (133 A+150 B+176 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (133 A+150 B+176 C) \sec (c+d x) \tan (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (67 A+90 B+80 C) \sec ^2(c+d x) \tan (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (3 A+10 B) \sqrt {a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {A (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.79
\[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (60 \sqrt {2} (133 A+150 B+176 C) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^5(c+d x)+(13313 A+11550 B+10480 C+12 (1273 A+1070 B+880 C) \cos (c+d x)+4 (3059 A+3450 B+3280 C) \cos (2 (c+d x))+2660 A \cos (3 (c+d x))+3000 B \cos (3 (c+d x))+3520 C \cos (3 (c+d x))+1995 A \cos (4 (c+d x))+2250 B \cos (4 (c+d x))+2640 C \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{15360 d}
\]
[In]
Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]
[Out]
(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(133*A + 150*B + 176*C)*ArcTanh[Sqrt
[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (13313*A + 11550*B + 10480*C + 12*(1273*A + 1070*B + 880*C)*Cos[c + d*x
] + 4*(3059*A + 3450*B + 3280*C)*Cos[2*(c + d*x)] + 2660*A*Cos[3*(c + d*x)] + 3000*B*Cos[3*(c + d*x)] + 3520*C
*Cos[3*(c + d*x)] + 1995*A*Cos[4*(c + d*x)] + 2250*B*Cos[4*(c + d*x)] + 2640*C*Cos[4*(c + d*x)])*Sin[(c + d*x)
/2]))/(15360*d)
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs. \(2(235)=470\).
Time = 16.99 (sec) , antiderivative size = 2683, normalized size of antiderivative =
10.20
| | |
| method | result | size |
| | |
| parts |
\(\text {Expression too large to display}\) |
\(2683\) |
| default |
\(\text {Expression too large to display}\) |
\(2879\) |
| | |
|
|
|
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[In]
int((a+cos(d*x+c)*a)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x,method=_RETURNVERBOSE)
[Out]
1/120*A*a^(1/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-63840*a*(ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2
))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))+ln(-4/(2*cos(1/2*d*x+1/2
*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)))*sin(1/2*d*x+1
/2*c)^10+31920*(2*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+5*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2
)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+5*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^
(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2*c
)^8-10640*(14*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+15*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a
*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+15*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1
/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2*c)^
6+1064*(128*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+75*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*c
os(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+75*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2
))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2*c)^4-
190*(316*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+105*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos
(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+105*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2)
)*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2*c)^2+1
1370*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+1995*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/
2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+1995*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*
(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)/(2*cos(1/2*d*x+1/2*c)-2^
(1/2))^5/(2*cos(1/2*d*x+1/2*c)+2^(1/2))^5/sin(1/2*d*x+1/2*c)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d+1/16*B*a^(1/2)*c
os(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(1200*2^(1/2)*a*(ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)
*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))+ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2)
)*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)))*sin(1/2*d*x+1/2*c)^8-240
0*(2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)
^(1/2)*a^(1/2)-2*a))*a+2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*si
n(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^6+200*(9*
2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1
/2)*a^(1/2)-2*a))*a+9*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin
(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+22*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^4+(-600
*2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(
1/2)*a^(1/2)-2*a))*a-600*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*
sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a-2920*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^2+
75*2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)
^(1/2)*a^(1/2)-2*a))*a+75*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+724*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))/(2*cos(1/2*d*x+1/2*c)
-2^(1/2))^4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))^4/sin(1/2*d*x+1/2*c)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d+1/12*
C*a^(1/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-264*2^(1/2)*a*(ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2
))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))+ln(-2/(2*cos(1/2*d*x+1/2
*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)))*sin(1/2*d*x+1
/2*c)^6+132*(3*2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d
*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+3*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)
+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+4*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+
1/2*c)^4-22*(9*2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d
*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+9*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)
+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+32*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x
+1/2*c)^2+33*2^(1/2)*ln(2/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+
1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a+33*2^(1/2)*ln(-2/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-
2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a+252*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))/(2*cos(1/2*
d*x+1/2*c)+2^(1/2))^3/(2*cos(1/2*d*x+1/2*c)-2^(1/2))^3/sin(1/2*d*x+1/2*c)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/
2)/d
Fricas [A] (verification not implemented)
none
Time = 0.41 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.96
\[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {15 \, {\left ({\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{6} + {\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (15 \, {\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{4} + 10 \, {\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (133 \, A + 150 \, B + 80 \, C\right )} a \cos \left (d x + c\right )^{2} + 48 \, {\left (19 \, A + 10 \, B\right )} a \cos \left (d x + c\right ) + 384 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}
\]
[In]
integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")
[Out]
1/7680*(15*((133*A + 150*B + 176*C)*a*cos(d*x + c)^6 + (133*A + 150*B + 176*C)*a*cos(d*x + c)^5)*sqrt(a)*log((
a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*sqrt(a*cos(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - 2)*sin(d*x + c) + 8
*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)) + 4*(15*(133*A + 150*B + 176*C)*a*cos(d*x + c)^4 + 10*(133*A + 150*B +
176*C)*a*cos(d*x + c)^3 + 8*(133*A + 150*B + 80*C)*a*cos(d*x + c)^2 + 48*(19*A + 10*B)*a*cos(d*x + c) + 384*A*
a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)
Sympy [F(-1)]
Timed out. \[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out}
\]
[In]
integrate((a+a*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)
[Out]
Timed out
Maxima [F(-1)]
Timed out. \[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out}
\]
[In]
integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="maxima")
[Out]
Timed out
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (235) = 470\).
Time = 0.85 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.86
\[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=-\frac {\sqrt {2} {\left (15 \, \sqrt {2} {\left (133 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 150 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 176 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) + \frac {4 \, {\left (31920 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 36000 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 42240 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 74480 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 84000 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 98560 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 68096 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76800 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 87040 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30020 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32760 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 34240 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5685 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5430 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5040 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}\right )} \sqrt {a}}{7680 \, d}
\]
[In]
integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="giac")
[Out]
-1/7680*sqrt(2)*(15*sqrt(2)*(133*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 150*B*a*sgn(cos(1/2*d*x + 1/2*c)) + 176*C*a*s
gn(cos(1/2*d*x + 1/2*c)))*log(abs(-2*sqrt(2) + 4*sin(1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(1/2*d*x + 1/2*c))
) + 4*(31920*A*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^9 + 36000*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/
2*d*x + 1/2*c)^9 + 42240*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^9 - 74480*A*a*sgn(cos(1/2*d*x + 1/
2*c))*sin(1/2*d*x + 1/2*c)^7 - 84000*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^7 - 98560*C*a*sgn(cos(
1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^7 + 68096*A*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^5 + 76800*
B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^5 + 87040*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c
)^5 - 30020*A*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 - 32760*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2
*d*x + 1/2*c)^3 - 34240*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)^3 + 5685*A*a*sgn(cos(1/2*d*x + 1/2*
c))*sin(1/2*d*x + 1/2*c) + 5430*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c) + 5040*C*a*sgn(cos(1/2*d*x
+ 1/2*c))*sin(1/2*d*x + 1/2*c))/(2*sin(1/2*d*x + 1/2*c)^2 - 1)^5)*sqrt(a)/d
Mupad [F(-1)]
Timed out. \[
\int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^6} \,d x
\]
[In]
int(((a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^6,x)
[Out]
int(((a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^6, x)