3.390 \(\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\)
Optimal. Leaf size=263 \[ \frac{a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{128 d}+\frac{a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
[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)
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Rubi [A] time = 0.762709, antiderivative size = 263, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used =
{3043, 2975, 2980, 2772, 2773, 206} \[ \frac{a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{128 d}+\frac{a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[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 3043
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])
Rule 2975
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*S
in[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 2980
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 2772
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 2773
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \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 (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 ) \operatorname{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) \tanh ^{-1}\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] time = 3.14001, size = 208, normalized size = 0.79 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt{a (\cos (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (12 (1273 A+1070 B+880 C) \cos (c+d x)+4 (3059 A+3450 B+3280 C) \cos (2 (c+d x))+2660 A \cos (3 (c+d x))+1995 A \cos (4 (c+d x))+13313 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+3520 C \cos (3 (c+d x))+2640 C \cos (4 (c+d x))+10480 C)+60 \sqrt{2} (133 A+150 B+176 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15360 d} \]
Antiderivative was successfully verified.
[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] time = 0.289, size = 2843, normalized size = 10.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((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]
1/120*a^(1/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-480*a*(133*A*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^
(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))+133*A*ln(4/(2*cos(1/
2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))+150*B
*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1
/2*c)+2*a))+150*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2
*d*x+1/2*c)^2)^(1/2)+2*a))+176*C*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^
2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))+176*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1
/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)))*sin(1/2*d*x+1/2*c)^10+240*(266*A*2^(1/2)*(a*sin(1/
2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+300*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+352*C*2^(1/2)*(a*sin(1/2*d*
x+1/2*c)^2)^(1/2)*a^(1/2)+665*A*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2
)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+665*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+
1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+750*B*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(
1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+750*B*ln(4/(2*cos(1/2*d*x+1/2
*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+880*C*ln(-4/
(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2
*a))*a+880*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+
1/2*c)^2)^(1/2)+2*a))*a)*sin(1/2*d*x+1/2*c)^8-80*(1862*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2100*B
*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2464*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+1995*A*l
n(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2
*c)+2*a))*a+1995*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/
2*d*x+1/2*c)^2)^(1/2)+2*a))*a+2250*B*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2
*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+2250*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/
2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+2640*C*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2
))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+2640*C*ln(4/(2*cos(1/2
*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a)*sin
(1/2*d*x+1/2*c)^6+8*(17024*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+19200*B*a^(1/2)*2^(1/2)*(a*sin(1/2
*d*x+1/2*c)^2)^(1/2)+21760*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+9975*A*ln(-4/(-2*cos(1/2*d*x+1/2*c
)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+9975*A*ln(4/(2
*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)
)*a+11250*B*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*co
s(1/2*d*x+1/2*c)+2*a))*a+11250*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(
1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+13200*C*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*
sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+13200*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(
a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a)*sin(1/2*d*x+1/2*c)^4-10*(
6004*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+6552*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+68
48*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+1995*A*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1
/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+1995*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1
/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+2250*B*ln(-4/(-2*cos
(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+
2250*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)
^2)^(1/2)+2*a))*a+2640*C*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)
-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+2640*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)
+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a)*sin(1/2*d*x+1/2*c)^2+11370*A*2^(1/2)*(a*sin(1/2*d*x+1
/2*c)^2)^(1/2)*a^(1/2)+1995*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2
)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+1995*A*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(
1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+10860*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^
(1/2)+2250*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+
1/2*c)^2)^(1/2)+2*a))*a+2250*B*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)
^(1/2)-a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*a+10080*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2640*C*ln(4
/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2
*a))*a+2640*C*ln(-4/(-2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)-a*2^(1/2)*
cos(1/2*d*x+1/2*c)+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
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
Fricas [A] time = 4.22125, size = 703, normalized size = 2.67 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
Giac [B] time = 3.83888, size = 1827, normalized size = 6.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
1/3840*(15*(133*A*a^(3/2) + 150*B*a^(3/2) + 176*C*a^(3/2))*log(abs((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(
1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sqrt(2) + 3))) - 15*(133*A*a^(3/2) + 150*B*a^(3/2) + 176*C*a^(3/2))*log(abs(
(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + a*(2*sqrt(2) - 3))) + 4*sqrt(2)*(1995*
(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^18*A*a^(5/2) + 2250*(sqrt(a)*tan(1/2*d*x +
1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^18*B*a^(5/2) + 2640*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1
/2*d*x + 1/2*c)^2 + a))^18*C*a^(5/2) - 38505*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a
))^16*A*a^(7/2) - 76110*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^16*B*a^(7/2) - 559
20*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^16*C*a^(7/2) + 561660*(sqrt(a)*tan(1/2*
d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*a^(9/2) + 737160*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(
a*tan(1/2*d*x + 1/2*c)^2 + a))^14*B*a^(9/2) + 582720*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*
c)^2 + a))^14*C*a^(9/2) - 2684100*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^12*A*a^(
11/2) - 3492600*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^12*B*a^(11/2) - 3395520*(s
qrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^12*C*a^(11/2) + 7371738*(sqrt(a)*tan(1/2*d*x
+ 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A*a^(13/2) + 9022860*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a
*tan(1/2*d*x + 1/2*c)^2 + a))^10*B*a^(13/2) + 9329760*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2
*c)^2 + a))^10*C*a^(13/2) - 6407470*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*A*a^
(15/2) - 7635300*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*B*a^(15/2) - 8110880*(s
qrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*C*a^(15/2) + 2176620*(sqrt(a)*tan(1/2*d*x
+ 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*A*a^(17/2) + 2614440*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*t
an(1/2*d*x + 1/2*c)^2 + a))^6*B*a^(17/2) + 2882880*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)
^2 + a))^6*C*a^(17/2) - 399860*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*A*a^(19/2
) - 460440*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*B*a^(19/2) - 498880*(sqrt(a)*
tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*C*a^(19/2) + 34035*(sqrt(a)*tan(1/2*d*x + 1/2*c)
- sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*A*a^(21/2) + 41850*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x
+ 1/2*c)^2 + a))^2*B*a^(21/2) + 42960*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*C*
a^(21/2) - 1201*A*a^(23/2) - 1470*B*a^(23/2) - 1520*C*a^(23/2))/((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/
2*d*x + 1/2*c)^2 + a))^4 - 6*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a + a^2)^5)
/d