3.400 \(\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^7(c+d x) \, dx\)
Optimal. Leaf size=311 \[ \frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^2 (115 A+156 B+120 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \tan (c+d x) \sec ^2(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a (5 A+12 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
[Out]
(a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(
1015*A + 1132*B + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Sec
[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Sec[c + d*x]^2*Tan[c +
d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3
*Tan[c + d*x])/(480*d) + (a*(5*A + 12*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(60*d) + (A*(
a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)
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Rubi [A] time = 0.966408, antiderivative size = 311, normalized size of antiderivative = 1.,
number of steps used = 8, 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} \[ \frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^2 (115 A+156 B+120 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \tan (c+d x) \sec ^2(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a (5 A+12 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
[Out]
(a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(
1015*A + 1132*B + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Sec
[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Sec[c + d*x]^2*Tan[c +
d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3
*Tan[c + d*x])/(480*d) + (a*(5*A + 12*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(60*d) + (A*(
a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*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))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (5 A+12 B)+\frac{1}{2} a (5 A+12 C) \cos (c+d x)\right ) \sec ^6(c+d x) \, dx}{6 a}\\ &=\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (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) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{30 a}\\ &=\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int \sqrt{a+a \cos (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) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a}\\ &=\frac{a^3 (545 A+628 B+680 C) \sec ^2(c+d x) \tan (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{384} \left (a^2 (1015 A+1132 B+1304 C)\right ) \int \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \, dx\\ &=\frac{a^3 (1015 A+1132 B+1304 C) \sec (c+d x) \tan (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (545 A+628 B+680 C) \sec ^2(c+d x) \tan (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{512} \left (a^2 (1015 A+1132 B+1304 C)\right ) \int \sqrt{a+a \cos (c+d x)} \sec ^2(c+d x) \, dx\\ &=\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1015 A+1132 B+1304 C) \sec (c+d x) \tan (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (545 A+628 B+680 C) \sec ^2(c+d x) \tan (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\left (a^2 (1015 A+1132 B+1304 C)\right ) \int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx}{1024}\\ &=\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1015 A+1132 B+1304 C) \sec (c+d x) \tan (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (545 A+628 B+680 C) \sec ^2(c+d x) \tan (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{\left (a^3 (1015 A+1132 B+1304 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 )}{512 d}\\ &=\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{512 d}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1015 A+1132 B+1304 C) \sec (c+d x) \tan (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (545 A+628 B+680 C) \sec ^2(c+d x) \tan (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (115 A+156 B+120 C) \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \tan (c+d x)}{480 d}+\frac{a (5 A+12 B) (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \tan (c+d x)}{60 d}+\frac{A (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 3.94586, size = 242, normalized size = 0.78 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \sqrt{a (\cos (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) ((321370 A+303048 B+283920 C) \cos (c+d x)+16 (8555 A+8444 B+7480 C) \cos (2 (c+d x))+108605 A \cos (3 (c+d x))+20300 A \cos (4 (c+d x))+15225 A \cos (5 (c+d x))+137060 A+121124 B \cos (3 (c+d x))+22640 B \cos (4 (c+d x))+16980 B \cos (5 (c+d x))+112464 B+127240 C \cos (3 (c+d x))+26080 C \cos (4 (c+d x))+19560 C \cos (5 (c+d x))+93600 C)+120 \sqrt{2} (1015 A+1132 B+1304 C) \cos ^6(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{122880 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
[Out]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^6*(120*Sqrt[2]*(1015*A + 1132*B + 1304*C)*ArcTan
h[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^6 + (137060*A + 112464*B + 93600*C + (321370*A + 303048*B + 283920*C)
*Cos[c + d*x] + 16*(8555*A + 8444*B + 7480*C)*Cos[2*(c + d*x)] + 108605*A*Cos[3*(c + d*x)] + 121124*B*Cos[3*(c
+ d*x)] + 127240*C*Cos[3*(c + d*x)] + 20300*A*Cos[4*(c + d*x)] + 22640*B*Cos[4*(c + d*x)] + 26080*C*Cos[4*(c
+ d*x)] + 15225*A*Cos[5*(c + d*x)] + 16980*B*Cos[5*(c + d*x)] + 19560*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(
122880*d)
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Maple [B] time = 0.411, size = 3316, normalized size = 10.7 \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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x)
[Out]
1/240*a^(3/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(960*a*(1015*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))+1015*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))+1132
*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))+1132*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))+1304*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))+1304*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)))*sin(1/2*d*x+1/2*c)^12-960*(1015*A*2^(1/2)*(a*
sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+1132*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+1304*C*2^(1/2)*(a*si
n(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+3045*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+3045*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+3396*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+3396*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+
3912*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+3912*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)*sin(1/2*d*x+1/2*c)^10+80*(34510*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2
)*a^(1/2)+38488*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+44336*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2
)*a^(1/2)+45675*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+45675*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+50940*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+50940*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+58680*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+5
8680*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)*sin(1/2*d*x+1/2*c)^8-96*(33495*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+37356*B*a
^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+42520*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+25375*A*l
n(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+25375*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+28300*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+28300*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+32600*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+32600*C*ln(-4/(-2*co
s(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
)*sin(1/2*d*x+1/2*c)^6+12*(162980*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+180304*B*a^(1/2)*2^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)+198560*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+76125*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+76125*
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+84900*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*si
n(1/2*d*x+1/2*c)^2)^(1/2)+2*a))*a+84900*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+97800*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+97800*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)*sin(1/2*d*x+1/
2*c)^4-20*(31897*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+34004*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c
)^2)^(1/2)+35176*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+9135*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+9135*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+10188*
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+10188*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+11736*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+11736*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)*sin(1/2*d*x+1/2*c)^2+15225*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+92430*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+15225*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+16980*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+88920*B*
a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+16980*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+19560*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+83760*C*2^(1/2)*(a*sin(
1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+19560*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)/(2*cos(1/2*d*x+1/2*c)-2^(1/2))^6/(2*cos(1/2*d*x+1/2*c)+2^(
1/2))^6/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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
Fricas [A] time = 4.25491, size = 810, normalized size = 2.6 \begin{align*} \frac{15 \,{\left ({\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} +{\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \,{\left (1015 \, A + 1132 \, B + 1304 \, 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} + 48 \,{\left (145 \, A + 116 \, B + 40 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \,{\left (35 \, A + 12 \, B\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, A a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{30720 \,{\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")
[Out]
1/30720*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)^7 + (1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)^6)*sq
rt(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*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)^5 + 10*(
1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)^4 + 8*(1015*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 48*(145*A + 11
6*B + 40*C)*a^2*cos(d*x + c)^2 + 128*(35*A + 12*B)*a^2*cos(d*x + c) + 1280*A*a^2)*sqrt(a*cos(d*x + c) + a)*sin
(d*x + c))/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)
________________________________________________________________________________________
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))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 4.81421, size = 2159, normalized size = 6.94 \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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")
[Out]
1/15360*(15*(1015*A*a^(5/2) + 1132*B*a^(5/2) + 1304*C*a^(5/2))*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*(1015*A*a^(5/2) + 1132*B*a^(5/2) + 1304*C*a^(5/2))*l
og(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)
*(15225*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^22*A*a^(7/2) + 16980*(sqrt(a)*tan(
1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^22*B*a^(7/2) + 19560*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sq
rt(a*tan(1/2*d*x + 1/2*c)^2 + a))^22*C*a^(7/2) - 502425*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1
/2*c)^2 + a))^20*A*a^(9/2) - 560340*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^20*B*a
^(9/2) - 645480*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^20*C*a^(9/2) + 6518495*(sq
rt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^18*A*a^(11/2) + 7963020*(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^(11/2) + 8467800*(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^(11/2) - 49683495*(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^(13/2) - 56336940*(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^(13/2) - 59757720*(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^(13/2) + 191286
330*(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^(15/2) + 219014472*(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^(15/2) + 244004880*(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^(15/2) - 418895130*(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^(17/2) - 474348232*(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^(17/2) - 531000080*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^12*C*a^(
17/2) + 374587230*(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^(19/2) + 42176911
2*(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^(19/2) + 473308080*(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^(19/2) - 154254030*(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^(21/2) - 174597720*(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^(21/2) - 198757680*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 +
a))^8*C*a^(21/2) + 35939005*(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^(23/2) +
40114980*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*B*a^(23/2) + 45352200*(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^(23/2) - 4649085*(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^(25/2) - 5273124*(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^(25/2) - 5884680*(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^(25/2) + 324435*(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^(27/2) + 36
7644*(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^(27/2) + 411000*(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^(27/2) - 9435*A*a^(29/2) - 10684*B*a^(29/2) - 11960
*C*a^(29/2))/((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)^6)/d