3.5.92 \(\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [492]

Optimal. Leaf size=187 \[ \frac {8 a^2 (63 A+57 B+47 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d} \]

[Out]

2/315*(63*A-18*B+22*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/9*C*sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c
)/d+2/21*(3*B+C)*(a+a*sec(d*x+c))^(5/2)*tan(d*x+c)/a/d+8/315*a^2*(63*A+57*B+47*C)*tan(d*x+c)/d/(a+a*sec(d*x+c)
)^(1/2)+2/315*a*(63*A+57*B+47*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

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Rubi [A]
time = 0.34, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {4173, 4095, 4086, 3878, 3877} \begin {gather*} \frac {8 a^2 (63 A+57 B+47 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (63 A-18 B+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac {2 a (63 A+57 B+47 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 (3 B+C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*(63*A + 57*B + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a
+ a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(31
5*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Sec[c + d*x])^
(5/2)*Tan[c + d*x])/(21*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 4095

Int[csc[(e_.) + (f_.)*(x_)]^2*(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 + 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*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x], x] /; Fr
eeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] &&  !LtQ[m, -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 ^2(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 {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C)+\frac {3}{2} a (3 B+C) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {4 \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {15}{4} a^2 (3 B+C)+\frac {1}{4} a^2 (63 A-18 B+22 C) \sec (c+d x)\right ) \, dx}{63 a^2}\\ &=\frac {2 (63 A-18 B+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {1}{105} (63 A+57 B+47 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {1}{315} (4 a (63 A+57 B+47 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {8 a^2 (63 A+57 B+47 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}\\ \end {align*}

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Mathematica [A]
time = 2.26, size = 152, normalized size = 0.81 \begin {gather*} \frac {a (693 A+702 B+752 C+(567 A+648 B+748 C) \cos (c+d x)+(882 A+858 B+748 C) \cos (2 (c+d x))+189 A \cos (3 (c+d x))+156 B \cos (3 (c+d x))+136 C \cos (3 (c+d x))+189 A \cos (4 (c+d x))+156 B \cos (4 (c+d x))+136 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 )}{630 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(693*A + 702*B + 752*C + (567*A + 648*B + 748*C)*Cos[c + d*x] + (882*A + 858*B + 748*C)*Cos[2*(c + d*x)] +
189*A*Cos[3*(c + d*x)] + 156*B*Cos[3*(c + d*x)] + 136*C*Cos[3*(c + d*x)] + 189*A*Cos[4*(c + d*x)] + 156*B*Cos[
4*(c + d*x)] + 136*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(630*d)

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Maple [A]
time = 14.38, size = 172, normalized size = 0.92

method	result	size
default	\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (378 A \left (\cos ^{4}\left (d x +c \right )\right )+312 B \left (\cos ^{4}\left (d x +c \right )\right )+272 C \left (\cos ^{4}\left (d x +c \right )\right )+189 A \left (\cos ^{3}\left (d x +c \right )\right )+156 B \left (\cos ^{3}\left (d x +c \right )\right )+136 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+117 B \left (\cos ^{2}\left (d x +c \right )\right )+102 C \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right )+85 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/d*(-1+cos(d*x+c))*(378*A*cos(d*x+c)^4+312*B*cos(d*x+c)^4+272*C*cos(d*x+c)^4+189*A*cos(d*x+c)^3+156*B*co
s(d*x+c)^3+136*C*cos(d*x+c)^3+63*A*cos(d*x+c)^2+117*B*cos(d*x+c)^2+102*C*cos(d*x+c)^2+45*B*cos(d*x+c)+85*C*cos
(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

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

[Out]

4/315*(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)*((A*a*d*cos(2*d*x + 2*c)^4
+ A*a*d*sin(2*d*x + 2*c)^4 + 4*A*a*d*cos(2*d*x + 2*c)^3 + 6*A*a*d*cos(2*d*x + 2*c)^2 + 4*A*a*d*cos(2*d*x + 2*
c) + A*a*d + 2*(A*a*d*cos(2*d*x + 2*c)^2 + 2*A*a*d*cos(2*d*x + 2*c) + A*a*d)*sin(2*d*x + 2*c)^2)*integrate((co
s(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(((cos(10*d*x + 10*c)*cos(2*d*x + 2*c) +
4*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 6*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 4*cos(4*d*x + 4*c)*cos(2*d*x + 2*
c) + cos(2*d*x + 2*c)^2 + sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 4*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 6*sin(6*
d*x + 6*c)*sin(2*d*x + 2*c) + 4*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(10*d*x + 10*c) + 4*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) +
6*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 4*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(10*d*x + 10*c)*sin(2*d*x + 2*c
) - 4*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 6*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 4*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(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c
) + 1)) - ((cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 4*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 6*cos(2*d*x + 2*c)*sin
(6*d*x + 6*c) + 4*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 4*cos(8*d*x + 8*c)
*sin(2*d*x + 2*c) - 6*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 4*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(10*d*x + 10*c)*cos(2*d*x + 2*c) + 4*cos(8*d*x + 8*c)*cos(2*d*x +
2*c) + 6*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 4*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(10
*d*x + 10*c)*sin(2*d*x + 2*c) + 4*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 6*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 4*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*
sin(3/2*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(10*d*x + 10*c)^2 + 16*(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 + 36*(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 + 16*(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(10*d*x + 10*c)^2 + 16*(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 + 36*(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 + 16*(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 + 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) + 6*(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) + 4*(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(10*d*x + 10*c) + 8*(cos(2*d*x
+ 2*c)^3 + cos(2*d*x + 2*c)*sin(2*d*x + 2*c)^2 + 6*(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) + 4*(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) + 12*(cos(2*d*x + 2*c)^3 + cos(2*d*x + 2*c)*si
n(2*d*x + 2*c)^2 + 4*(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*c
os(2*d*x + 2*c)^2 + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 8*(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 + 4*(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) + 6*(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) + 4*(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(10*d*x + 10*c) + 8*(sin(2*d*x + 2*c)^3 + 6*(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) + 4*(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) + 12*(sin(2*d*x + 2*c)^3 +
4*(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(6*d*x + 6*c) + 8*(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(3/2*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)...

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Fricas [A]
time = 1.85, size = 135, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (2 \, {\left (189 \, A + 156 \, B + 136 \, C\right )} a \cos \left (d x + c\right )^{4} + {\left (189 \, A + 156 \, B + 136 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 39 \, B + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, B + 17 \, C\right )} a \cos \left (d x + c\right ) + 35 \, C a\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)^2*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

2/315*(2*(189*A + 156*B + 136*C)*a*cos(d*x + c)^4 + (189*A + 156*B + 136*C)*a*cos(d*x + c)^3 + 3*(21*A + 39*B
+ 34*C)*a*cos(d*x + c)^2 + 5*(9*B + 17*C)*a*cos(d*x + c) + 35*C*a)*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 {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sec(c + d*x) + 1))**(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**2, x)

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Giac [A]
time = 1.46, size = 324, normalized size = 1.73 \begin {gather*} \frac {4 \, {\left ({\left ({\left ({\left (2 \, \sqrt {2} {\left (63 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 57 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 47 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} {\left (63 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 57 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 47 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 63 \, \sqrt {2} {\left (17 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \sqrt {2} {\left (9 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 7 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \sqrt {2} {\left (A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\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)^2*(a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

4/315*((((2*sqrt(2)*(63*A*a^6*sgn(cos(d*x + c)) + 57*B*a^6*sgn(cos(d*x + c)) + 47*C*a^6*sgn(cos(d*x + c)))*tan
(1/2*d*x + 1/2*c)^2 - 9*sqrt(2)*(63*A*a^6*sgn(cos(d*x + c)) + 57*B*a^6*sgn(cos(d*x + c)) + 47*C*a^6*sgn(cos(d*
x + c))))*tan(1/2*d*x + 1/2*c)^2 + 63*sqrt(2)*(17*A*a^6*sgn(cos(d*x + c)) + 13*B*a^6*sgn(cos(d*x + c)) + 13*C*
a^6*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 105*sqrt(2)*(9*A*a^6*sgn(cos(d*x + c)) + 7*B*a^6*sgn(cos(d*x
+ c)) + 5*C*a^6*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 315*sqrt(2)*(A*a^6*sgn(cos(d*x + c)) + B*a^6*sgn(
cos(d*x + c)) + C*a^6*sgn(cos(d*x + c))))*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 = 12.34, size = 709, normalized size = 3.79 \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\,a\,4{}\mathrm {i}}{5\,d}+\frac {a\,\left (3\,A+6\,B+4\,C\right )\,4{}\mathrm {i}}{5\,d}+\frac {a\,\left (3\,B+C\right )\,16{}\mathrm {i}}{105\,d}\right )-\frac {a\,\left (3\,A+2\,B\right )\,4{}\mathrm {i}}{5\,d}+\frac {a\,\left (A+4\,B+12\,C\right )\,4{}\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\,\left (3\,A+2\,B\right )\,4{}\mathrm {i}}{9\,d}-\frac {A\,a\,4{}\mathrm {i}}{9\,d}+\frac {a\,\left (5\,A+6\,B+4\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a\,\left (7\,A+8\,B+12\,C\right )\,4{}\mathrm {i}}{9\,d}\right )-\frac {a\,\left (3\,A+2\,B\right )\,4{}\mathrm {i}}{9\,d}+\frac {A\,a\,4{}\mathrm {i}}{9\,d}-\frac {a\,\left (5\,A+6\,B+4\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a\,\left (7\,A+8\,B+12\,C\right )\,4{}\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\,a\,4{}\mathrm {i}}{3\,d}-\frac {a\,\left (21\,A+39\,B+34\,C\right )\,8{}\mathrm {i}}{315\,d}\right )+\frac {a\,\left (3\,A+2\,B\right )\,4{}\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\,4{}\mathrm {i}}{7\,d}+\frac {C\,a\,32{}\mathrm {i}}{63\,d}-\frac {a\,\left (2\,A+3\,B+2\,C\right )\,8{}\mathrm {i}}{7\,d}+\frac {a\,\left (3\,A+2\,B+8\,C\right )\,4{}\mathrm {i}}{7\,d}\right )+\frac {a\,\left (3\,A+2\,B\right )\,4{}\mathrm {i}}{7\,d}+\frac {a\,\left (A-8\,C\right )\,4{}\mathrm {i}}{7\,d}-\frac {a\,\left (2\,A+3\,B+6\,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 {a\,{\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 (189\,A+156\,B+136\,C\right )\,4{}\mathrm {i}}{315\,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))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^2,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*(3*A + 6*B + 4*C)*4i)/(
5*d) - (A*a*4i)/(5*d) + (a*(3*B + C)*16i)/(105*d)) - (a*(3*A + 2*B)*4i)/(5*d) + (a*(A + 4*B + 12*C)*4i)/(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)*((a*(3*A + 2*B)*4i)/(9*d) - (A*a*4i)/(9*d) + (a*(5*A + 6*B + 4*C)*4i)/(9*d) - (
a*(7*A + 8*B + 12*C)*4i)/(9*d)) - (a*(3*A + 2*B)*4i)/(9*d) + (A*a*4i)/(9*d) - (a*(5*A + 6*B + 4*C)*4i)/(9*d) +
(a*(7*A + 8*B + 12*C)*4i)/(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*a*4i)/(3*d) - (a*(21*A + 39*B + 34*C)*8i)/
(315*d)) + (a*(3*A + 2*B)*4i)/(3*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) + ((a + a/(exp(- c*1
i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((A*a*4i)/(7*d) + (C*a*32i)/(63*d) - (a*(2*A
+ 3*B + 2*C)*8i)/(7*d) + (a*(3*A + 2*B + 8*C)*4i)/(7*d)) + (a*(3*A + 2*B)*4i)/(7*d) + (a*(A - 8*C)*4i)/(7*d) -
(a*(2*A + 3*B + 6*C)*8i)/(7*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^3) - (a*exp(c*1i + d*x*1i
)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(189*A + 156*B + 136*C)*4i)/(315*d*(exp(c*1i +
d*x*1i) + 1))

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