3.509 \(\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=311 \[ \frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a^2 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x) \cos (c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.894864, 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 =
{4086, 4017, 4015, 3805, 3774, 203} \[ \frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a^2 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x) \cos (c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]
Antiderivative was successfully verified.
[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 4086
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])
Rule 4017
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 4015
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 3805
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 3774
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 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \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 \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 ) \operatorname{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) \tan ^{-1}\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 [A] time = 3.42134, size = 217, normalized size = 0.7 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (15 \sqrt{2} (1015 A+1132 B+1304 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+\left (\sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (2 (8085 A+7748 B+7240 C) \cos (c+d x)+4 (1575 A+1324 B+920 C) \cos (2 (c+d x))+2140 A \cos (3 (c+d x))+560 A \cos (4 (c+d x))+80 A \cos (5 (c+d x))+20965 A+1392 B \cos (3 (c+d x))+192 B \cos (4 (c+d x))+22084 B+480 C \cos (3 (c+d x))+23240 C)\right )}{15360 d} \]
Antiderivative was successfully verified.
[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*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(1015*A + 1132*B + 1304*C)*ArcSin[Sqrt[2]*Sin[(c
+ d*x)/2]]*Sqrt[Cos[c + d*x]] + (20965*A + 22084*B + 23240*C + 2*(8085*A + 7748*B + 7240*C)*Cos[c + d*x] + 4*(
1575*A + 1324*B + 920*C)*Cos[2*(c + d*x)] + 2140*A*Cos[3*(c + d*x)] + 1392*B*Cos[3*(c + d*x)] + 480*C*Cos[3*(c
+ d*x)] + 560*A*Cos[4*(c + d*x)] + 192*B*Cos[4*(c + d*x)] + 80*A*Cos[5*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(
3*(c + d*x))/2])))/(15360*d)
________________________________________________________________________________________
Maple [B] time = 0.404, size = 1654, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
1/491520/d*a^2*(195600*C*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(
1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+76125*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))
-158720*A*cos(d*x+c)^10-74240*A*cos(d*x+c)^9-363520*C*cos(d*x+c)^8-223232*B*cos(d*x+c)^9+974400*A*cos(d*x+c)^6
+16980*B*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c
)/(cos(d*x+c)+1))^(11/2)*sin(d*x+c)+15225*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2)*arctanh(1/2*2^(1/2)*
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*sin(d*x+c)-98304*B*cos(d*x+c)^11-258048*B*cos(d*x+
c)^10-129920*A*cos(d*x+c)^8-324800*A*cos(d*x+c)^7-362240*B*cos(d*x+c)^7+1086720*B*cos(d*x+c)^6+97800*C*cos(d*x
+c)*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*sin(d*x+c)/cos(d*x+c))+15225*A*cos(d*x+c)^5*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*a
rctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+19560*C*cos(d*x+c)^5*sin(d*x+c)
*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*
x+c)/cos(d*x+c))+76125*A*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(
1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+97800*C*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)*(-2*c
os(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c
))+152250*A*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+195600*C*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+152250*A*
cos(d*x+c)^2*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+16980*B*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d
*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2)*cos(d*x+c)^5*sin(d*x+c)+84900*B*arctanh(1/2*2^
(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2
)*cos(d*x+c)^4*sin(d*x+c)+169800*B*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x
+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+169800*B*arctanh(1/2*2^(1/2)*(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2)*cos(d*x+c)
^2*sin(d*x+c)+84900*B*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1/2)*cos(d*x+c)*sin(d*x+c)+19560*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)*2^(1
/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*sin(d*x+c)-81920*A*cos(d*x
+c)^12-204800*A*cos(d*x+c)^11-122880*C*cos(d*x+c)^10-348160*C*cos(d*x+c)^9+1251840*C*cos(d*x+c)^6-144896*B*cos
(d*x+c)^8-417280*C*cos(d*x+c)^7)*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)/cos(d*x+c)^5/sin(d*x+c)
________________________________________________________________________________________
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(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
________________________________________________________________________________________
Fricas [A] time = 1.336, size = 1507, normalized size = 4.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)] 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(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
________________________________________________________________________________________
Giac [B] time = 9.33477, size = 2989, normalized size = 9.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
-1/15360*(15*(1015*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1132*B*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1304*C*sqrt(-a)*
a^2*sgn(cos(d*x + c)))*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*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1132*B*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1304*C*sq
rt(-a)*a^2*sgn(cos(d*x + c)))*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)*(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*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 19560*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)
^2 + a))^22*C*sqrt(-a)*a^3*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 6518495*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a
*tan(1/2*d*x + 1/2*c)^2 + a))^18*A*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 7963020*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - s
qrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^18*B*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^5*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) + 191286330
*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*sqrt(-a)*a^7*sgn(cos(d*x + c)) + 2
19014472*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*B*sqrt(-a)*a^7*sgn(cos(d*x +
c)) + 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*sqrt(-a)*a^7*sgn(c
os(d*x + c)) - 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*sqrt(-a)*a
^8*sgn(cos(d*x + c)) - 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*sq
rt(-a)*a^8*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^8*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^9*sgn(cos(d*x + c)) + 421769112*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x
+ 1/2*c)^2 + a))^10*B*sqrt(-a)*a^9*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^9*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^10*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^10*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^10*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^11*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^11*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^11*sgn(cos(d*x + c)) - 46
49085*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*A*sqrt(-a)*a^12*sgn(cos(d*x + c)
) - 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*sqrt(-a)*a^12*sgn(cos(d*
x + c)) - 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*sqrt(-a)*a^12*sgn(
cos(d*x + c)) + 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*sqrt(-a)*a^13
*sgn(cos(d*x + c)) + 367644*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*B*sqrt(-a)
*a^13*sgn(cos(d*x + c)) + 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*sqr
t(-a)*a^13*sgn(cos(d*x + c)) - 9435*A*sqrt(-a)*a^14*sgn(cos(d*x + c)) - 10684*B*sqrt(-a)*a^14*sgn(cos(d*x + c)
) - 11960*C*sqrt(-a)*a^14*sgn(cos(d*x + c)))/((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