3.6.2 \(\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [502]

Optimal. Leaf size=184 \[ \frac {64 a^3 (21 A+15 B+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d} \]

[Out]

2/105*a*(21*A+15*B+13*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/63*(9*B-2*C)*(a+a*sec(d*x+c))^(5/2)*tan(d*x+c)/
d+2/9*C*(a+a*sec(d*x+c))^(7/2)*tan(d*x+c)/a/d+64/315*a^3*(21*A+15*B+13*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+
16/315*a^2*(21*A+15*B+13*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

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Rubi [A]
time = 0.24, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4167, 4086, 3878, 3877} \begin {gather*} \frac {64 a^3 (21 A+15 B+13 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (21 A+15 B+13 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+15 B+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 (9 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(64*a^3*(21*A + 15*B + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqr
t[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x
])/(105*d) + (2*(9*B - 2*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*
Tan[c + d*x])/(9*a*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 3878

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

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 4167

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

Rubi steps

\begin {align*} \int \sec (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 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {2 \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (9 A+7 C)+\frac {1}{2} a (9 B-2 C) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{21} (21 A+15 B+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (21 A+15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{105} (8 a (21 A+15 B+13 C)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{315} \left (32 a^2 (21 A+15 B+13 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 (21 A+15 B+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end {align*}

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Mathematica [A]
time = 2.55, size = 156, normalized size = 0.85 \begin {gather*} \frac {a^2 (2961 A+2790 B+2908 C+2 (882 A+1215 B+1396 C) \cos (c+d x)+4 (966 A+870 B+803 C) \cos (2 (c+d x))+588 A \cos (3 (c+d x))+690 B \cos (3 (c+d x))+584 C \cos (3 (c+d x))+903 A \cos (4 (c+d x))+690 B \cos (4 (c+d x))+584 C \cos (4 (c+d x))) \sec ^4(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(2961*A + 2790*B + 2908*C + 2*(882*A + 1215*B + 1396*C)*Cos[c + d*x] + 4*(966*A + 870*B + 803*C)*Cos[2*(c
+ d*x)] + 588*A*Cos[3*(c + d*x)] + 690*B*Cos[3*(c + d*x)] + 584*C*Cos[3*(c + d*x)] + 903*A*Cos[4*(c + d*x)] +
690*B*Cos[4*(c + d*x)] + 584*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/
(1260*d)

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Maple [A]
time = 13.95, size = 174, normalized size = 0.95

method	result	size
default	\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (903 A \left (\cos ^{4}\left (d x +c \right )\right )+690 B \left (\cos ^{4}\left (d x +c \right )\right )+584 C \left (\cos ^{4}\left (d x +c \right )\right )+294 A \left (\cos ^{3}\left (d x +c \right )\right )+345 B \left (\cos ^{3}\left (d x +c \right )\right )+292 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+180 B \left (\cos ^{2}\left (d x +c \right )\right )+219 C \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right )+130 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/d*(-1+cos(d*x+c))*(903*A*cos(d*x+c)^4+690*B*cos(d*x+c)^4+584*C*cos(d*x+c)^4+294*A*cos(d*x+c)^3+345*B*co
s(d*x+c)^3+292*C*cos(d*x+c)^3+63*A*cos(d*x+c)^2+180*B*cos(d*x+c)^2+219*C*cos(d*x+c)^2+45*B*cos(d*x+c)+130*C*co
s(d*x+c)+35*C)*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)*a^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(315*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(((5*A + 2*B)*a^2*d*cos(2*
d*x + 2*c)^4 + (5*A + 2*B)*a^2*d*sin(2*d*x + 2*c)^4 + 4*(5*A + 2*B)*a^2*d*cos(2*d*x + 2*c)^3 + 6*(5*A + 2*B)*a
^2*d*cos(2*d*x + 2*c)^2 + 4*(5*A + 2*B)*a^2*d*cos(2*d*x + 2*c) + (5*A + 2*B)*a^2*d + 2*((5*A + 2*B)*a^2*d*cos(
2*d*x + 2*c)^2 + 2*(5*A + 2*B)*a^2*d*cos(2*d*x + 2*c) + (5*A + 2*B)*a^2*d)*sin(2*d*x + 2*c)^2)*integrate((((co
s(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) +
cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 3*sin(4*d*x + 4
*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x +
2*c)*sin(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x
+ 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 3*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(9/2*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos
(2*d*x + 2*c)*sin(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - c
os(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 3*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*
cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6*d*x + 6*c)
*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*
c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 3*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(9/2*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((cos(
2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(8
*d*x + 8*c)^2 + 9*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c)^2 + 9*(c
os(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c)^2 + 2*cos(2*d*x + 2*c)^3 + (
cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(8*d*x + 8*c)^2 + 9*(cos(2*d*x + 2*c)^2 +
sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(6*d*x + 6*c)^2 + 9*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2
+ 2*cos(2*d*x + 2*c) + 1)*sin(4*d*x + 4*c)^2 + (2*cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c
)^2 + 2*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2
+ 2*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)
+ 1)*cos(4*d*x + 4*c) + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 6*(cos(2*d*x + 2*c)^3 + co
s(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(4
*d*x + 4*c) + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x +
2*c)*sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(4*d*x + 4*c) + cos(2*d*x + 2*c)^2 + 2*(
sin(2*d*x + 2*c)^3 + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(6*d*x + 6*c) + 3
*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(4*d*x + 4*c) + (cos(2*d*x + 2*c)^2 + 2
*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 6*(sin(2*d*x + 2*c)^3 + 3*(cos(2*d*x + 2*c)^2 + si
n(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(4*d*x + 4*c) + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*si
n(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(sin(2*d*x + 2*c)^3 + (cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2
*d*x + 2*c))*sin(4*d*x + 4*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (cos(2*d*x + 2*c)^
4 + sin(2*d*x + 2*c)^4 + (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c)^2
+ 9*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c)^2 + 9*(cos(2*d*x + 2*
c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c)^2 + 2*cos(2*d*x + 2*c)^3 + (cos(2*d*x + 2
*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(8*d*x + 8*c)^2 + 9*(cos(2*d*x + 2*c)^2 + sin(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(6*d*x + 6*c)^2 + 9*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)*sin(4*d*x + 4*c)^2 + (2*cos(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 2*(cos(
2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*
x + 2*c) + 1)*cos(6*d*x + 6*c) + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(4*d*
x + 4*c) + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c
)*sin(2*d*x + 2*c)^2 + 3*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) +
2*cos(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 6*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*
x + 2*c)^2 + 2*cos(2*d*x + 2*c)^2 + cos(2*d*x +...

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Fricas [A]
time = 3.49, size = 144, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left ({\left (903 \, A + 690 \, B + 584 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (294 \, A + 345 \, B + 292 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 60 \, B + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, B + 26 \, C\right )} a^{2} \cos \left (d x + c\right ) + 35 \, 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 )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((903*A + 690*B + 584*C)*a^2*cos(d*x + c)^4 + (294*A + 345*B + 292*C)*a^2*cos(d*x + c)^3 + 3*(21*A + 60*
B + 73*C)*a^2*cos(d*x + c)^2 + 5*(9*B + 26*C)*a^2*cos(d*x + c) + 35*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x +
c))*sin(d*x + c)/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sec(c + d*x) + 1))**(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (164) = 328\).
time = 1.69, size = 348, normalized size = 1.89 \begin {gather*} \frac {8 \, {\left (315 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 315 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 315 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (1050 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 840 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 630 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (1323 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 945 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 819 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 4 \, {\left (189 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 135 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 117 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 2 \, {\left (21 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 15 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, \sqrt {2} C a^{7} \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 )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \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)*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

8/315*(315*sqrt(2)*A*a^7*sgn(cos(d*x + c)) + 315*sqrt(2)*B*a^7*sgn(cos(d*x + c)) + 315*sqrt(2)*C*a^7*sgn(cos(d
*x + c)) - (1050*sqrt(2)*A*a^7*sgn(cos(d*x + c)) + 840*sqrt(2)*B*a^7*sgn(cos(d*x + c)) + 630*sqrt(2)*C*a^7*sgn
(cos(d*x + c)) - (1323*sqrt(2)*A*a^7*sgn(cos(d*x + c)) + 945*sqrt(2)*B*a^7*sgn(cos(d*x + c)) + 819*sqrt(2)*C*a
^7*sgn(cos(d*x + c)) - 4*(189*sqrt(2)*A*a^7*sgn(cos(d*x + c)) + 135*sqrt(2)*B*a^7*sgn(cos(d*x + c)) + 117*sqrt
(2)*C*a^7*sgn(cos(d*x + c)) - 2*(21*sqrt(2)*A*a^7*sgn(cos(d*x + c)) + 15*sqrt(2)*B*a^7*sgn(cos(d*x + c)) + 13*
sqrt(2)*C*a^7*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)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^4*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*
d)

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Mupad [B]
time = 16.80, size = 885, normalized size = 4.81 \begin {gather*} \frac {\left (\frac {A\,a^2\,2{}\mathrm {i}}{d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (903\,A+690\,B+584\,C\right )\,2{}\mathrm {i}}{315\,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}}}}{{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1}-\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 (3\,A+4\,B+4\,C\right )\,10{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (11\,A+10\,B+20\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,2{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (9\,A-16\,C\right )\,2{}\mathrm {i}}{63\,d}\right )+\frac {A\,a^2\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,A+2\,B-16\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (11\,A+10\,B+4\,C\right )\,2{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (15\,A+20\,B+36\,C\right )\,2{}\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 {\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 (189\,A+240\,B+292\,C\right )\,2{}\mathrm {i}}{315\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,2{}\mathrm {i}}{3\,d}\right )+\frac {A\,a^2\,2{}\mathrm {i}}{3\,d}-\frac {a^2\,\left (9\,A+10\,B+4\,C\right )\,2{}\mathrm {i}}{3\,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 )}-\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\,2{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (6\,A+5\,B+2\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (10\,A+11\,B+10\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (13\,A+15\,B+20\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,2{}\mathrm {i}}{9\,d}\right )-\frac {A\,a^2\,2{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (6\,A+5\,B+2\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (10\,A+11\,B+10\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (13\,A+15\,B+20\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,2{}\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 (5\,A+9\,B+10\,C\right )\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (24\,B-21\,A+32\,C\right )\,2{}\mathrm {i}}{105\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,2{}\mathrm {i}}{5\,d}\right )+\frac {A\,a^2\,2{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (5\,A+5\,B+2\,C\right )\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (5\,A+10\,B+32\,C\right )\,2{}\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} \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),x)

[Out]

(((A*a^2*2i)/d - (a^2*exp(c*1i + d*x*1i)*(903*A + 690*B + 584*C)*2i)/(315*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) - ((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*(3*A + 4*B + 4*C)*10i)/(7*d) - (a^2*(11*A + 10*B + 20*C)*2i)/(7*d) - (a^2
*(5*A + 2*B)*2i)/(7*d) + (a^2*(9*A - 16*C)*2i)/(63*d)) + (A*a^2*2i)/(7*d) - (a^2*(5*A + 2*B - 16*C)*2i)/(7*d)
- (a^2*(11*A + 10*B + 4*C)*2i)/(7*d) + (a^2*(15*A + 20*B + 36*C)*2i)/(7*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*
2i + d*x*2i) + 1)^3) - ((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*(189*A + 240*B + 292*C)*2i)/(315*d) - (a^2*(5*A + 2*B)*2i)/(3*d)) + (A*a^2*2i)/(3*d) - (a^2*(9*A + 10*B + 4*
C)*2i)/(3*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1
i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((A*a^2*2i)/(9*d) - (a^2*(6*A + 5*B + 2*C)*4i)/(9*d) - (a^2*(10*A +
11*B + 10*C)*4i)/(9*d) + (a^2*(13*A + 15*B + 20*C)*4i)/(9*d) + (a^2*(5*A + 2*B)*2i)/(9*d)) - (A*a^2*2i)/(9*d)
+ (a^2*(6*A + 5*B + 2*C)*4i)/(9*d) + (a^2*(10*A + 11*B + 10*C)*4i)/(9*d) - (a^2*(13*A + 15*B + 20*C)*4i)/(9*d)
- (a^2*(5*A + 2*B)*2i)/(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*(5*A + 9*B + 10*C)*4i)/(5*d) + (a^2*(24*B -
21*A + 32*C)*2i)/(105*d) - (a^2*(5*A + 2*B)*2i)/(5*d)) + (A*a^2*2i)/(5*d) - (a^2*(5*A + 5*B + 2*C)*4i)/(5*d)
+ (a^2*(5*A + 10*B + 32*C)*2i)/(5*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2)

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