3.4.66 \(\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [366]
Optimal. Leaf size=189 \[ \frac {2 a^2 (39 B+34 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (9 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 B+34 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 B+34 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d} \]
[Out]
2/105*(39*B+34*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/45*a^2*(39*B+34*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)
+2/63*a^2*(9*B+10*C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-4/315*a*(39*B+34*C)*(a+a*sec(d*x+c))^(1/
2)*tan(d*x+c)/d+2/9*a*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 = 0.38, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4157, 4103,
4101, 3885, 4086, 3877} \begin {gather*} \frac {2 a^2 (9 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (39 B+34 C) \tan (c+d x)}{45 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (39 B+34 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac {4 a (39 B+34 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{9 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*a^2*(39*B + 34*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(9*B + 10*C)*Sec[c + d*x]^3*Tan[c
+ d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(39*B + 34*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) +
(2*a*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(39*B + 34*C)*(a + a*Sec[c + d*x])^(3/
2)*Tan[c + d*x])/(105*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 4157
Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(
x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.), x_Symbol] :> Dist[1/b^2, Int[(a + b*Csc[e + f*x])
^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} (B+C \sec (c+d x)) \, dx\\ &=\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2}{9} \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{2} a (3 B+2 C)+\frac {1}{2} a (9 B+10 C) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (9 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {1}{21} (a (39 B+34 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (9 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 B+34 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {1}{105} (2 (39 B+34 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\\ &=\frac {2 a^2 (9 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 B+34 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 B+34 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {1}{45} (a (39 B+34 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (39 B+34 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (9 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 B+34 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 B+34 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 100, normalized size = 0.53 \begin {gather*} \frac {2 a^2 \left (8 (39 B+34 C)+4 (39 B+34 C) \sec (c+d x)+3 (39 B+34 C) \sec ^2(c+d x)+5 (9 B+17 C) \sec ^3(c+d x)+35 C \sec ^4(c+d x)\right ) \tan (c+d x)}{315 d \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*a^2*(8*(39*B + 34*C) + 4*(39*B + 34*C)*Sec[c + d*x] + 3*(39*B + 34*C)*Sec[c + d*x]^2 + 5*(9*B + 17*C)*Sec[c
+ d*x]^3 + 35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])
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Maple [A]
time = 14.15, size = 139, normalized size = 0.74
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (312 B \left (\cos ^{4}\left (d x +c \right )\right )+272 C \left (\cos ^{4}\left (d x +c \right )\right )+156 B \left (\cos ^{3}\left (d x +c \right )\right )+136 C \left (\cos ^{3}\left (d x +c \right )\right )+117 B \left (\cos ^{2}\left (d x +c \right )\right )+102 C \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right )+85 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\) |
\(139\) |
| | |
|
|
|
|
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)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
-2/315/d*(-1+cos(d*x+c))*(312*B*cos(d*x+c)^4+272*C*cos(d*x+c)^4+156*B*cos(d*x+c)^3+136*C*cos(d*x+c)^3+117*B*co
s(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)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
8/315*(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)*(((3*B + 2*C)*a*d*cos(2*d*
x + 2*c)^4 + (3*B + 2*C)*a*d*sin(2*d*x + 2*c)^4 + 4*(3*B + 2*C)*a*d*cos(2*d*x + 2*c)^3 + 6*(3*B + 2*C)*a*d*cos
(2*d*x + 2*c)^2 + 4*(3*B + 2*C)*a*d*cos(2*d*x + 2*c) + (3*B + 2*C)*a*d + 2*((3*B + 2*C)*a*d*cos(2*d*x + 2*c)^2
+ 2*(3*B + 2*C)*a*d*cos(2*d*x + 2*c) + (3*B + 2*C)*a*d)*sin(2*d*x + 2*c)^2)*integrate((cos(2*d*x + 2*c)^2 + s
in(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(7/2*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)*s
in(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(7/2*arcta
n2(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*co
s(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(7/2*arctan2(sin(2*d*x + 2*c), c
os(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(7/2*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)*s
in(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)*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(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) + 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*a...
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Fricas [A]
time = 2.10, size = 127, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left (8 \, {\left (39 \, B + 34 \, C\right )} a \cos \left (d x + c\right )^{4} + 4 \, {\left (39 \, B + 34 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (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)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
2/315*(8*(39*B + 34*C)*a*cos(d*x + c)^4 + 4*(39*B + 34*C)*a*cos(d*x + c)^3 + 3*(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 (B + C \sec {\left (c + d x \right )}\right ) \sec ^{3}{\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)*(B*sec(d*x+c)+C*sec(d*x+c)**2),x)
[Out]
Integral((a*(sec(c + d*x) + 1))**(3/2)*(B + C*sec(c + d*x))*sec(c + d*x)**3, x)
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Giac [A]
time = 1.28, size = 258, normalized size = 1.37 \begin {gather*} \frac {4 \, {\left ({\left ({\left ({\left (2 \, \sqrt {2} {\left (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 (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} + 819 \, \sqrt {2} {\left (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 )^{2} - 105 \, \sqrt {2} {\left (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 (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)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
4/315*((((2*sqrt(2)*(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)*(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 + 819*sqrt(2)*(B*a^6*sgn(
cos(d*x + c)) + C*a^6*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 105*sqrt(2)*(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)*(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 = 11.40, size = 596, normalized size = 3.15 \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\,\left (3\,B+2\,C\right )\,8{}\mathrm {i}}{7\,d}+\frac {C\,a\,32{}\mathrm {i}}{63\,d}+\frac {a\,\left (B+4\,C\right )\,8{}\mathrm {i}}{7\,d}\right )+\frac {B\,a\,8{}\mathrm {i}}{7\,d}-\frac {C\,a\,32{}\mathrm {i}}{7\,d}-\frac {a\,\left (B+2\,C\right )\,24{}\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\,\left (2\,B+3\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a\,\left (3\,B+2\,C\right )\,8{}\mathrm {i}}{9\,d}+\frac {B\,a\,8{}\mathrm {i}}{9\,d}\right )+\frac {a\,\left (2\,B+3\,C\right )\,16{}\mathrm {i}}{9\,d}-\frac {a\,\left (3\,B+2\,C\right )\,8{}\mathrm {i}}{9\,d}-\frac {B\,a\,8{}\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 {\left (\frac {B\,a\,8{}\mathrm {i}}{3\,d}-\frac {a\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (39\,B+34\,C\right )\,8{}\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}}}}{\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\,\left (3\,B+2\,C\right )\,8{}\mathrm {i}}{5\,d}+\frac {a\,\left (3\,B+C\right )\,16{}\mathrm {i}}{105\,d}\right )-\frac {B\,a\,8{}\mathrm {i}}{5\,d}+\frac {a\,\left (B+3\,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 {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 (39\,B+34\,C\right )\,16{}\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(((B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(3/2))/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)*((C*a*32i)/(63*d) - (a*(3*B
+ 2*C)*8i)/(7*d) + (a*(B + 4*C)*8i)/(7*d)) + (B*a*8i)/(7*d) - (C*a*32i)/(7*d) - (a*(B + 2*C)*24i)/(7*d)))/((e
xp(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*(3*B + 2*C)*8i)/(9*d) - (a*(2*B + 3*C)*16i)/(9*d) + (B*a*8i)/(9*d)) + (a*(2*B +
3*C)*16i)/(9*d) - (a*(3*B + 2*C)*8i)/(9*d) - (B*a*8i)/(9*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) +
1)^4) + (((B*a*8i)/(3*d) - (a*exp(c*1i + d*x*1i)*(39*B + 34*C)*8i)/(315*d))*(a + a/(exp(- c*1i - d*x*1i)/2 + e
xp(c*1i + d*x*1i)/2))^(1/2))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) + ((a + a/(exp(- c*1i - d*x*1
i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a*(3*B + 2*C)*8i)/(5*d) + (a*(3*B + C)*16i)/(105*d))
- (B*a*8i)/(5*d) + (a*(B + 3*C)*16i)/(5*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2) - (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)*(39*B + 34*C)*16i)/(315*d*(exp(c*1
i + d*x*1i) + 1))
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