3.2.56 \(\int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} (A+C \sec ^2(c+d x)) \, dx\) [156]
Optimal. Leaf size=180 \[ \frac {2 a (21 A+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d} \]
[Out]
2/105*(21*A+16*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/a/d+2/45*a*(21*A+16*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)
+2/63*a*C*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-4/315*(21*A+16*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)
/d+2/9*C*sec(d*x+c)^3*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4174, 4101,
3885, 4086, 3877} \begin {gather*} \frac {2 (21 A+16 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac {4 (21 A+16 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+16 C) \tan (c+d x)}{45 d \sqrt {a \sec (c+d x)+a}}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{9 d}+\frac {2 a C \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]
[Out]
(2*a*(21*A + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*S
qrt[a + a*Sec[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]
^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1
05*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 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 4174
Int[((A_.) + 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*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, 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) \sqrt {a+a \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{2} a (3 A+2 C)+\frac {1}{2} a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 a C \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {1}{21} (21 A+16 C) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a C \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {(2 (21 A+16 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}{105 a}\\ &=\frac {2 a C \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {1}{45} (21 A+16 C) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (21 A+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.06, size = 122, normalized size = 0.68 \begin {gather*} \frac {(189 A+214 C+2 (63 A+88 C) \cos (c+d x)+11 (21 A+16 C) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+32 C \cos (3 (c+d x))+42 A \cos (4 (c+d x))+32 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 )}{315 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]
[Out]
((189*A + 214*C + 2*(63*A + 88*C)*Cos[c + d*x] + 11*(21*A + 16*C)*Cos[2*(c + d*x)] + 42*A*Cos[3*(c + d*x)] + 3
2*C*Cos[3*(c + d*x)] + 42*A*Cos[4*(c + d*x)] + 32*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])
]*Tan[(c + d*x)/2])/(315*d)
________________________________________________________________________________________
Maple [A]
time = 10.14, size = 129, normalized size = 0.72
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (168 A \left (\cos ^{4}\left (d x +c \right )\right )+128 C \left (\cos ^{4}\left (d x +c \right )\right )+84 A \left (\cos ^{3}\left (d x +c \right )\right )+64 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+48 C \left (\cos ^{2}\left (d x +c \right )\right )+40 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\) |
\(129\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)^3*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-2/315/d*(-1+cos(d*x+c))*(168*A*cos(d*x+c)^4+128*C*cos(d*x+c)^4+84*A*cos(d*x+c)^3+64*C*cos(d*x+c)^3+63*A*cos(d
*x+c)^2+48*C*cos(d*x+c)^2+40*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)
________________________________________________________________________________________
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)^3*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),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)*((A*d*cos(2*d*x + 2*c)^4 +
A*d*sin(2*d*x + 2*c)^4 + 4*A*d*cos(2*d*x + 2*c)^3 + 6*A*d*cos(2*d*x + 2*c)^2 + 4*A*d*cos(2*d*x + 2*c) + 2*(A*
d*cos(2*d*x + 2*c)^2 + 2*A*d*cos(2*d*x + 2*c) + A*d)*sin(2*d*x + 2*c)^2 + A*d)*integrate((((cos(12*d*x + 12*c)
*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*
x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(
2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6
*c)*sin(2*d*x + 2*c) + 5*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(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 10*co
s(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) -
cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*sin(2*d*x +
2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(7/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(
12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x +
2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*
d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5
*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(12*d*x + 12*c)
*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*
x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(
2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6
*c)*sin(2*d*x + 2*c) + 5*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(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(5*cos(10*d*x + 10*c)
+ 10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(
12*d*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(
10*d*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*co
s(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*co
s(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*si
n(10*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d
*x + 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2
*d*x + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x + 10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin
(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*
x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x +
2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(5*cos(10*d*x + 10*c) + 10*cos(8*d*x
+ 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^
2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c)
+ 25*cos(10*d*x + 10*c)^2 + 20*(10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c)
+ 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*d*x + 6*c)^
2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10*d*x + 10*c
) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + si
n(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*si
n(10*d*x + 10*c) + 25*sin(10*d*x + 10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*
sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*
sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/
2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c) + 1)^(1/4)), x) + 2*((A + 2*C)*d*cos(2*d*x + 2*c)^4 + (A + 2*C)*d*sin(2*d*x + 2*c)^4 + 4*(A + 2*C)*d*cos
(2*d*x + 2*c)^3 + 6*(A + 2*C)*d*cos(2*d*x + 2*c)^2 + 4*(A + 2*C)*d*cos(2*d*x + 2*c) + 2*((A + 2*C)*d*cos(2*d*x
+ 2*c)^2 + 2*(A + 2*C)*d*cos(2*d*x + 2*c) + (A...
________________________________________________________________________________________
Fricas [A]
time = 3.06, size = 116, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (8 \, {\left (21 \, A + 16 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A + 16 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 16 \, C\right )} \cos \left (d x + c\right )^{2} + 40 \, C \cos \left (d x + c\right ) + 35 \, C\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)^3*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
2/315*(8*(21*A + 16*C)*cos(d*x + c)^4 + 4*(21*A + 16*C)*cos(d*x + c)^3 + 3*(21*A + 16*C)*cos(d*x + c)^2 + 40*C
*cos(d*x + c) + 35*C)*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)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (A + C \sec ^{2}{\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)**3*(A+C*sec(d*x+c)**2)*(a+a*sec(d*x+c))**(1/2),x)
[Out]
Integral(sqrt(a*(sec(c + d*x) + 1))*(A + C*sec(c + d*x)**2)*sec(c + d*x)**3, x)
________________________________________________________________________________________
Giac [A]
time = 1.07, size = 256, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left ({\left ({\left ({\left (\sqrt {2} {\left (147 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 107 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \sqrt {2} {\left (14 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 9 \, C a^{5} \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} + 882 \, \sqrt {2} {\left (A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{5} \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} - 420 \, \sqrt {2} {\left (2 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{5} \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^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{5} \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)^3*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
[Out]
2/315*((((sqrt(2)*(147*A*a^5*sgn(cos(d*x + c)) + 107*C*a^5*sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)^2 - 36*sqrt
(2)*(14*A*a^5*sgn(cos(d*x + c)) + 9*C*a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 882*sqrt(2)*(A*a^5*sgn(
cos(d*x + c)) + C*a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 420*sqrt(2)*(2*A*a^5*sgn(cos(d*x + c)) + C*
a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 315*sqrt(2)*(A*a^5*sgn(cos(d*x + c)) + C*a^5*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)
________________________________________________________________________________________
Mupad [B]
time = 11.23, size = 535, normalized size = 2.97 \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\,8{}\mathrm {i}}{7\,d}+\frac {C\,32{}\mathrm {i}}{63\,d}+\frac {\left (72\,A+288\,C\right )\,1{}\mathrm {i}}{63\,d}\right )+\frac {A\,8{}\mathrm {i}}{7\,d}-\frac {C\,32{}\mathrm {i}}{7\,d}-\frac {\left (72\,A+288\,C\right )\,1{}\mathrm {i}}{63\,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 (-\frac {A\,16{}\mathrm {i}}{9\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,16{}\mathrm {i}}{9\,d}-\frac {\left (16\,A+32\,C\right )\,1{}\mathrm {i}}{9\,d}\right )+\frac {\left (16\,A+32\,C\right )\,1{}\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 {A\,8{}\mathrm {i}}{3\,d}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (168\,A+128\,C\right )\,1{}\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\,8{}\mathrm {i}}{5\,d}-\frac {C\,32{}\mathrm {i}}{105\,d}\right )-\frac {A\,8{}\mathrm {i}}{5\,d}+\frac {C\,32{}\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 {{\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 (336\,A+256\,C\right )\,1{}\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 + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(1/2))/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)*((C*32i)/(63*d) - (A*8i)/(7
*d) + ((72*A + 288*C)*1i)/(63*d)) + (A*8i)/(7*d) - (C*32i)/(7*d) - ((72*A + 288*C)*1i)/(63*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*16i)/(9*d) - ((16*A + 32*C)*1i)/(9*d)) - (A*16i)/(9*d) + ((16*A + 32*C)*1i)/(9*d)))/((exp(c
*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^4) + (((A*8i)/(3*d) - (exp(c*1i + d*x*1i)*(168*A + 128*C)*1i)/(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)*(exp(c*2i + d*x*
2i) + 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*8i)/(5*d) -
(C*32i)/(105*d)) - (A*8i)/(5*d) + (C*32i)/(5*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2) - (ex
p(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(336*A + 256*C)*1i)/(315*d*(exp
(c*1i + d*x*1i) + 1))
________________________________________________________________________________________