3.499 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=263 \[ \frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a^2 (67 A+90 B+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a (3 A+10 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
[Out]
(a^(3/2)*(133*A + 150*B + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*
A + 150*B + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Cos[c + d*x]*
Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d
*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A
*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)
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Rubi [A] time = 0.672605, 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 =
{4086, 4017, 4015, 3805, 3774, 203} \[ \frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a^2 (67 A+90 B+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a (3 A+10 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(a^(3/2)*(133*A + 150*B + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*
A + 150*B + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Cos[c + d*x]*
Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d
*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A
*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*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 ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{2} a (3 A+10 B)+\frac{5}{2} a (A+2 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{\int \cos ^3(c+d x) \sqrt{a+a \sec (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) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (67 A+90 B+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{96} (a (133 A+150 B+176 C)) \int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (133 A+150 B+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (67 A+90 B+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{128} (a (133 A+150 B+176 C)) \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (133 A+150 B+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (67 A+90 B+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{256} (a (133 A+150 B+176 C)) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (133 A+150 B+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (67 A+90 B+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (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 \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}\\ &=\frac{a^{3/2} (133 A+150 B+176 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}+\frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (133 A+150 B+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (67 A+90 B+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (3 A+10 B) \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.67023, size = 182, normalized size = 0.69 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (15 \sqrt{2} (133 A+150 B+176 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+\left (\sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (2 (1007 A+930 B+880 C) \cos (c+d x)+4 (181 A+150 B+80 C) \cos (2 (c+d x))+228 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+2671 A+120 B \cos (3 (c+d x))+2850 B+2960 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(133*A + 150*B + 176*C)*ArcSin[Sqrt[2]*Sin[(c + d*x
)/2]]*Sqrt[Cos[c + d*x]] + (2671*A + 2850*B + 2960*C + 2*(1007*A + 930*B + 880*C)*Cos[c + d*x] + 4*(181*A + 15
0*B + 80*C)*Cos[2*(c + d*x)] + 228*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[
(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)
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Maple [B] time = 0.365, size = 1379, normalized size = 5.2 \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)^5*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-1/61440/d*a*(13500*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*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)*cos(d*x+c)^2*2^(1/2)+15840*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arct
anh(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)*cos(d*x+c)^2*2^(1/2)+12
288*A*cos(d*x+c)^10+16896*A*cos(d*x+c)^9+20480*C*cos(d*x+c)^8+15360*B*cos(d*x+c)^9+9000*B*(-2*cos(d*x+c)/(cos(
d*x+c)+1))^(9/2)*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)*co
s(d*x+c)*2^(1/2)+10560*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*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)*cos(d*x+c)*2^(1/2)+1995*A*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))^(9/2)*cos(d*x+c)^4*sin(d*x+c)+79
80*A*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)/(c
os(d*x+c)+1))^(9/2)*cos(d*x+c)^3*sin(d*x+c)+11970*A*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))^(9/2)*cos(d*x+c)^2*sin(d*x+c)+7980*A*2^(1/2)*arct
anh(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))^(9/
2)*cos(d*x+c)*sin(d*x+c)+2250*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*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)*cos(d*x+c)^4*2^(1/2)+2640*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(
9/2)*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)*cos(d*x+c)^4*2
^(1/2)+9000*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*si
n(d*x+c)/cos(d*x+c))*sin(d*x+c)*cos(d*x+c)^3*2^(1/2)+10560*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*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)*cos(d*x+c)^3*2^(1/2)+21280*A*co
s(d*x+c)^6-63840*A*cos(d*x+c)^5-72000*B*cos(d*x+c)^5-84480*C*cos(d*x+c)^5+4864*A*cos(d*x+c)^8+8512*A*cos(d*x+c
)^7+9600*B*cos(d*x+c)^7+24000*B*cos(d*x+c)^6+28160*C*cos(d*x+c)^6+2250*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*
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)+2640*C*(-2*
cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*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)/c
os(d*x+c))*sin(d*x+c)+1995*A*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))^(9/2)*sin(d*x+c)+23040*B*cos(d*x+c)^8+35840*C*cos(d*x+c)^7)*(a*(cos(d*x+
c)+1)/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)
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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)^5*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [A] time = 1.31934, size = 1295, normalized size = 4.92 \begin{align*} \left [\frac{15 \,{\left ({\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right ) +{\left (133 \, A + 150 \, B + 176 \, C\right )} a\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \,{\left (384 \, A a \cos \left (d x + c\right )^{5} + 48 \,{\left (19 \, A + 10 \, B\right )} a \cos \left (d x + c\right )^{4} + 8 \,{\left (133 \, A + 150 \, B + 80 \, C\right )} a \cos \left (d x + c\right )^{3} + 10 \,{\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \,{\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3840 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{15 \,{\left ({\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right ) +{\left (133 \, A + 150 \, B + 176 \, C\right )} a\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) -{\left (384 \, A a \cos \left (d x + c\right )^{5} + 48 \,{\left (19 \, A + 10 \, B\right )} a \cos \left (d x + c\right )^{4} + 8 \,{\left (133 \, A + 150 \, B + 80 \, C\right )} a \cos \left (d x + c\right )^{3} + 10 \,{\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \,{\left (133 \, A + 150 \, B + 176 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1920 \,{\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/3840*(15*((133*A + 150*B + 176*C)*a*cos(d*x + c) + (133*A + 150*B + 176*C)*a)*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*(384*A*a*cos(d*x + c)^5 + 48*(19*A + 10*B)*a*cos(d*x + c)^4 + 8*(133*A + 150*B + 80*C)*a*co
s(d*x + c)^3 + 10*(133*A + 150*B + 176*C)*a*cos(d*x + c)^2 + 15*(133*A + 150*B + 176*C)*a*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), -1/1920*(15*((133*A + 150*B + 176*C)*a*c
os(d*x + c) + (133*A + 150*B + 176*C)*a)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(
sqrt(a)*sin(d*x + c))) - (384*A*a*cos(d*x + c)^5 + 48*(19*A + 10*B)*a*cos(d*x + c)^4 + 8*(133*A + 150*B + 80*C
)*a*cos(d*x + c)^3 + 10*(133*A + 150*B + 176*C)*a*cos(d*x + c)^2 + 15*(133*A + 150*B + 176*C)*a*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)**5*(a+a*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 8.00739, size = 2519, normalized size = 9.58 \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)^5*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
-1/3840*(15*(133*A*sqrt(-a)*a*sgn(cos(d*x + c)) + 150*B*sqrt(-a)*a*sgn(cos(d*x + c)) + 176*C*sqrt(-a)*a*sgn(co
s(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*(133*A*sqrt(-a)*a*sgn(cos(d*x + c)) + 150*B*sqrt(-a)*a*sgn(cos(d*x + c)) + 176*C*sqrt(-a)*a*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)*(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*sqrt(-a)*a^2*s
gn(cos(d*x + c)) + 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*sqrt(-a)*a^
2*sgn(cos(d*x + c)) + 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*sqrt(-a)
*a^2*sgn(cos(d*x + c)) - 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*sqrt
(-a)*a^3*sgn(cos(d*x + c)) - 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*
sqrt(-a)*a^3*sgn(cos(d*x + c)) - 55920*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^1
6*C*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^5*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^5*sgn(cos(d*x + c)) - 3395520*(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^5*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^6*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^7*sgn(cos(d*x + c)) - 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*sqrt(-a)*a^7*sgn(cos(d*x + c)) - 8110880*(sqr
t(-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^7*sgn(cos(d*x + c)) + 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*sqrt(-a)*a^8*sgn(cos(d*x + c)) + 26
14440*(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^8*sgn(cos(d*x + c))
+ 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*sqrt(-a)*a^8*sgn(cos(d*x
+ c)) - 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*sqrt(-a)*a^9*sgn(cos(
d*x + c)) - 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*sqrt(-a)*a^9*sgn(
cos(d*x + c)) - 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*sqrt(-a)*a^9*
sgn(cos(d*x + c)) + 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*sqrt(-a)*a
^10*sgn(cos(d*x + c)) + 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*sqrt(-
a)*a^10*sgn(cos(d*x + c)) + 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*sq
rt(-a)*a^10*sgn(cos(d*x + c)) - 1201*A*sqrt(-a)*a^11*sgn(cos(d*x + c)) - 1470*B*sqrt(-a)*a^11*sgn(cos(d*x + c)
) - 1520*C*sqrt(-a)*a^11*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)^5)/d