3.5.100 \(\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [500]

Optimal. Leaf size=294 \[ \frac {2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2717 A+2522 B+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d} \]

[Out]

2/15015*a*(10439*A+9230*B+8368*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/143*a*(13*B+5*C)*sec(d*x+c)^3*(a+a*sec
(d*x+c))^(3/2)*tan(d*x+c)/d+2/13*C*sec(d*x+c)^3*(a+a*sec(d*x+c))^(5/2)*tan(d*x+c)/d+2/6435*a^3*(10439*A+9230*B
+8368*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/9009*a^3*(2717*A+2522*B+2224*C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*
sec(d*x+c))^(1/2)-4/45045*a^2*(10439*A+9230*B+8368*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d+2/1287*a^2*(143*A+18
2*B+136*C)*sec(d*x+c)^3*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

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Rubi [A]
time = 0.63, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4173, 4103, 4101, 3885, 4086, 3877} \begin {gather*} \frac {2 a^3 (2717 A+2522 B+2224 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (143 A+182 B+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d}-\frac {4 a^2 (10439 A+9230 B+8368 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{45045 d}+\frac {2 a (10439 A+9230 B+8368 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a (13 B+5 C) \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(2*a^3*(10439*A + 9230*B + 8368*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2717*A + 2522*B +
2224*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*S
qrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec
[c + d*x]]*Tan[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/
(15015*d) + (2*a*(13*B + 5*C)*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d
*x]^3*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)

Rule 3877

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(Cot[e + f*x]/(
f*Sqrt[a + b*Csc[e + f*x]])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 3885

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*(
(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*
(b*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 4086

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Simp[(-B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*B*m + A*b*(m + 1))/(b
*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B
, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] &&  !LtQ[m, -2^(-1)]

Rule 4101

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[-2*b*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])
), x] + Dist[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x], x]
/; FreeQ[{a, b, d, e, f, A, B, n}, 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 4103

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m +
n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d
*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] &&
NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1]

Rule 4173

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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(
(d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^
n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A
, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)] &&  !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]

Rubi steps

\begin {align*} \int \sec ^3(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 {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {2 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (13 A+6 C)+\frac {1}{2} a (13 B+5 C) \sec (c+d x)\right ) \, dx}{13 a}\\ &=\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {4 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (143 A+78 B+96 C)+\frac {1}{4} a^2 (143 A+182 B+136 C) \sec (c+d x)\right ) \, dx}{143 a}\\ &=\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {8 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{8} a^3 (715 A+598 B+560 C)+\frac {1}{8} a^3 (2717 A+2522 B+2224 C) \sec (c+d x)\right ) \, dx}{1287 a}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {(2 a (10439 A+9230 B+8368 C)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx}{15015}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{6435}\\ &=\frac {2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2717 A+2522 B+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}\\ \end {align*}

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Mathematica [A]
time = 2.24, size = 222, normalized size = 0.76 \begin {gather*} \frac {a^2 (322751 A+325910 B+343612 C+70 (4576 A+5083 B+5552 C) \cos (c+d x)+14 (32747 A+31850 B+30334 C) \cos (2 (c+d x))+141570 A \cos (3 (c+d x))+138450 B \cos (3 (c+d x))+125520 C \cos (3 (c+d x))+156585 A \cos (4 (c+d x))+138450 B \cos (4 (c+d x))+125520 C \cos (4 (c+d x))+20878 A \cos (5 (c+d x))+18460 B \cos (5 (c+d x))+16736 C \cos (5 (c+d x))+20878 A \cos (6 (c+d x))+18460 B \cos (6 (c+d x))+16736 C \cos (6 (c+d x))) \sec ^6(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{180180 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^2*(322751*A + 325910*B + 343612*C + 70*(4576*A + 5083*B + 5552*C)*Cos[c + d*x] + 14*(32747*A + 31850*B + 30
334*C)*Cos[2*(c + d*x)] + 141570*A*Cos[3*(c + d*x)] + 138450*B*Cos[3*(c + d*x)] + 125520*C*Cos[3*(c + d*x)] +
156585*A*Cos[4*(c + d*x)] + 138450*B*Cos[4*(c + d*x)] + 125520*C*Cos[4*(c + d*x)] + 20878*A*Cos[5*(c + d*x)] +
18460*B*Cos[5*(c + d*x)] + 16736*C*Cos[5*(c + d*x)] + 20878*A*Cos[6*(c + d*x)] + 18460*B*Cos[6*(c + d*x)] + 1
6736*C*Cos[6*(c + d*x)])*Sec[c + d*x]^6*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(180180*d)

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Maple [A]
time = 15.07, size = 240, normalized size = 0.82

method	result	size
default	\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (83512 A \left (\cos ^{6}\left (d x +c \right )\right )+73840 B \left (\cos ^{6}\left (d x +c \right )\right )+66944 C \left (\cos ^{6}\left (d x +c \right )\right )+41756 A \left (\cos ^{5}\left (d x +c \right )\right )+36920 B \left (\cos ^{5}\left (d x +c \right )\right )+33472 C \left (\cos ^{5}\left (d x +c \right )\right )+31317 A \left (\cos ^{4}\left (d x +c \right )\right )+27690 B \left (\cos ^{4}\left (d x +c \right )\right )+25104 C \left (\cos ^{4}\left (d x +c \right )\right )+18590 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+20920 C \left (\cos ^{3}\left (d x +c \right )\right )+5005 A \left (\cos ^{2}\left (d x +c \right )\right )+14560 B \left (\cos ^{2}\left (d x +c \right )\right )+18305 C \left (\cos ^{2}\left (d x +c \right )\right )+4095 B \cos \left (d x +c \right )+11970 C \cos \left (d x +c \right )+3465 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{45045 d \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-2/45045/d*(-1+cos(d*x+c))*(83512*A*cos(d*x+c)^6+73840*B*cos(d*x+c)^6+66944*C*cos(d*x+c)^6+41756*A*cos(d*x+c)^
5+36920*B*cos(d*x+c)^5+33472*C*cos(d*x+c)^5+31317*A*cos(d*x+c)^4+27690*B*cos(d*x+c)^4+25104*C*cos(d*x+c)^4+185
90*A*cos(d*x+c)^3+23075*B*cos(d*x+c)^3+20920*C*cos(d*x+c)^3+5005*A*cos(d*x+c)^2+14560*B*cos(d*x+c)^2+18305*C*c
os(d*x+c)^2+4095*B*cos(d*x+c)+11970*C*cos(d*x+c)+3465*C)*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^6/sin(
d*x+c)*a^2

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(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

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Fricas [A]
time = 2.73, size = 192, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (8 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (3718 \, A + 4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (143 \, A + 416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

2/45045*(8*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c)^6 + 4*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c)^5 +
3*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c)^4 + 5*(3718*A + 4615*B + 4184*C)*a^2*cos(d*x + c)^3 + 35*(143*
A + 416*B + 523*C)*a^2*cos(d*x + c)^2 + 315*(13*B + 38*C)*a^2*cos(d*x + c) + 3465*C*a^2)*sqrt((a*cos(d*x + c)
+ a)/cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3*(a+a*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [A]
time = 1.67, size = 472, normalized size = 1.61 \begin {gather*} \frac {8 \, {\left (45045 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 45045 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 45045 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (180180 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 150150 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 120120 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (342342 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 300300 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 294294 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (391248 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 356070 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 310596 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (265837 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 232375 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 212069 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 4 \, {\left (24167 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 21125 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 19279 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 2 \, {\left (1859 \, \sqrt {2} A a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1625 \, \sqrt {2} B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, \sqrt {2} C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{45045 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

8/45045*(45045*sqrt(2)*A*a^9*sgn(cos(d*x + c)) + 45045*sqrt(2)*B*a^9*sgn(cos(d*x + c)) + 45045*sqrt(2)*C*a^9*s
gn(cos(d*x + c)) - (180180*sqrt(2)*A*a^9*sgn(cos(d*x + c)) + 150150*sqrt(2)*B*a^9*sgn(cos(d*x + c)) + 120120*s
qrt(2)*C*a^9*sgn(cos(d*x + c)) - (342342*sqrt(2)*A*a^9*sgn(cos(d*x + c)) + 300300*sqrt(2)*B*a^9*sgn(cos(d*x +
c)) + 294294*sqrt(2)*C*a^9*sgn(cos(d*x + c)) - (391248*sqrt(2)*A*a^9*sgn(cos(d*x + c)) + 356070*sqrt(2)*B*a^9*
sgn(cos(d*x + c)) + 310596*sqrt(2)*C*a^9*sgn(cos(d*x + c)) - (265837*sqrt(2)*A*a^9*sgn(cos(d*x + c)) + 232375*
sqrt(2)*B*a^9*sgn(cos(d*x + c)) + 212069*sqrt(2)*C*a^9*sgn(cos(d*x + c)) - 4*(24167*sqrt(2)*A*a^9*sgn(cos(d*x
+ c)) + 21125*sqrt(2)*B*a^9*sgn(cos(d*x + c)) + 19279*sqrt(2)*C*a^9*sgn(cos(d*x + c)) - 2*(1859*sqrt(2)*A*a^9*
sgn(cos(d*x + c)) + 1625*sqrt(2)*B*a^9*sgn(cos(d*x + c)) + 1483*sqrt(2)*C*a^9*sgn(cos(d*x + c)))*tan(1/2*d*x +
1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan
(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^6*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a
)*d)

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Mupad [B]
time = 17.37, size = 1201, normalized size = 4.09 \begin {gather*} -\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (-\frac {a^2\,\left (A-16\,C\right )\,8{}\mathrm {i}}{11\,d}+\frac {C\,a^2\,128{}\mathrm {i}}{143\,d}-\frac {a^2\,\left (3\,A+4\,B+4\,C\right )\,40{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (11\,A+10\,B+20\,C\right )\,8{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{11\,d}\right )-\frac {A\,a^2\,8{}\mathrm {i}}{11\,d}-\frac {C\,a^2\,128{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (5\,A+2\,B-16\,C\right )\,8{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (11\,A+10\,B+4\,C\right )\,8{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (15\,A+20\,B+36\,C\right )\,8{}\mathrm {i}}{11\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (-{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a^2\,\left (A-8\,B\right )\,8{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,A+9\,B+10\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (13\,B-6\,C\right )\,64{}\mathrm {i}}{1287\,d}\right )+\frac {A\,a^2\,8{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,A+5\,B+2\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (5\,A+10\,B+32\,C\right )\,8{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (B+6\,C\right )\,64{}\mathrm {i}}{9\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a^2\,\left (403\,B-572\,A+1046\,C\right )\,16{}\mathrm {i}}{15015\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{5\,d}\right )+\frac {A\,a^2\,8{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (4\,A+5\,B+2\,C\right )\,16{}\mathrm {i}}{5\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,a^2\,8{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (6\,A+5\,B+2\,C\right )\,16{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (10\,A+11\,B+10\,C\right )\,16{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (13\,A+15\,B+20\,C\right )\,16{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{13\,d}\right )-\frac {A\,a^2\,8{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (6\,A+5\,B+2\,C\right )\,16{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (10\,A+11\,B+10\,C\right )\,16{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (13\,A+15\,B+20\,C\right )\,16{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{13\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a^2\,\left (5\,A+16\,B+20\,C\right )\,8{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (143\,A+650\,B+811\,C\right )\,32{}\mathrm {i}}{9009\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{7\,d}\right )+\frac {A\,a^2\,8{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (A-7\,C\right )\,32{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (9\,A+10\,B+4\,C\right )\,8{}\mathrm {i}}{7\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {\left (\frac {A\,a^2\,8{}\mathrm {i}}{3\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (10439\,A+9230\,B+8368\,C\right )\,8{}\mathrm {i}}{45045\,d}\right )\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (10439\,A+9230\,B+8368\,C\right )\,16{}\mathrm {i}}{45045\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^3,x)

[Out]

((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((A*a^2*8i)/(13*d) - (a^2*(
6*A + 5*B + 2*C)*16i)/(13*d) - (a^2*(10*A + 11*B + 10*C)*16i)/(13*d) + (a^2*(13*A + 15*B + 20*C)*16i)/(13*d) +
(a^2*(5*A + 2*B)*8i)/(13*d)) - (A*a^2*8i)/(13*d) + (a^2*(6*A + 5*B + 2*C)*16i)/(13*d) + (a^2*(10*A + 11*B + 1
0*C)*16i)/(13*d) - (a^2*(13*A + 15*B + 20*C)*16i)/(13*d) - (a^2*(5*A + 2*B)*8i)/(13*d)))/((exp(c*1i + d*x*1i)
+ 1)*(exp(c*2i + d*x*2i) + 1)^6) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((A*a^2*8i)/
(9*d) - exp(c*1i + d*x*1i)*((a^2*(A - 8*B)*8i)/(9*d) - (a^2*(5*A + 9*B + 10*C)*16i)/(9*d) + (a^2*(5*A + 2*B)*8
i)/(9*d) - (a^2*(13*B - 6*C)*64i)/(1287*d)) - (a^2*(5*A + 5*B + 2*C)*16i)/(9*d) + (a^2*(5*A + 10*B + 32*C)*8i)
/(9*d) + (a^2*(B + 6*C)*64i)/(9*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^4) - ((a + a/(exp(- c*
1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a^2*(403*B - 572*A + 1046*C)*16i)/(15015*d
) - (a^2*(5*A + 2*B)*8i)/(5*d)) + (A*a^2*8i)/(5*d) - (a^2*(4*A + 5*B + 2*C)*16i)/(5*d)))/((exp(c*1i + d*x*1i)
+ 1)*(exp(c*2i + d*x*2i) + 1)^2) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i +
d*x*1i)*((C*a^2*128i)/(143*d) - (a^2*(A - 16*C)*8i)/(11*d) - (a^2*(3*A + 4*B + 4*C)*40i)/(11*d) + (a^2*(11*A +
10*B + 20*C)*8i)/(11*d) + (a^2*(5*A + 2*B)*8i)/(11*d)) - (A*a^2*8i)/(11*d) - (C*a^2*128i)/(11*d) + (a^2*(5*A
+ 2*B - 16*C)*8i)/(11*d) + (a^2*(11*A + 10*B + 4*C)*8i)/(11*d) - (a^2*(15*A + 20*B + 36*C)*8i)/(11*d)))/((exp(
c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^5) + ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/
2)*(exp(c*1i + d*x*1i)*((a^2*(5*A + 16*B + 20*C)*8i)/(7*d) + (a^2*(143*A + 650*B + 811*C)*32i)/(9009*d) - (a^2
*(5*A + 2*B)*8i)/(7*d)) + (A*a^2*8i)/(7*d) - (a^2*(A - 7*C)*32i)/(7*d) - (a^2*(9*A + 10*B + 4*C)*8i)/(7*d)))/(
(exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^3) + (((A*a^2*8i)/(3*d) - (a^2*exp(c*1i + d*x*1i)*(10439*A +
9230*B + 8368*C)*8i)/(45045*d))*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2))/((exp(c*1i + d
*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) - (a^2*exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*
x*1i)/2))^(1/2)*(10439*A + 9230*B + 8368*C)*16i)/(45045*d*(exp(c*1i + d*x*1i) + 1))

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