3.5.82 \(\int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [482]

Optimal. Leaf size=239 \[ \frac {4 a (99 A+88 B+80 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (99 A+88 B+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d} \]

[Out]

4/1155*(99*A+88*B+80*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/a/d+4/495*a*(99*A+88*B+80*C)*tan(d*x+c)/d/(a+a*sec(d
*x+c))^(1/2)+2/693*a*(99*A+88*B+80*C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/99*a*(11*B+C)*sec(d*x
+c)^4*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-8/3465*(99*A+88*B+80*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d+2/11*C*s
ec(d*x+c)^4*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

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Rubi [A]
time = 0.38, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4173, 4101, 3888, 3885, 4086, 3877} \begin {gather*} \frac {2 a (99 A+88 B+80 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {4 (99 A+88 B+80 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac {8 (99 A+88 B+80 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {4 a (99 A+88 B+80 C) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (11 B+C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {2 C \tan (c+d x) \sec ^4(c+d x) \sqrt {a \sec (c+d x)+a}}{11 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(4*a*(99*A + 88*B + 80*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Sec[c + d
*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt
[a + a*Sec[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*C*Sec[c +
d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Sec[c + d*x])^(3/2)*Tan
[c + d*x])/(1155*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 3888

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

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 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 ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {2 \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{2} a (11 A+8 C)+\frac {1}{2} a (11 B+C) \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {1}{99} (99 A+88 B+80 C) \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (99 A+88 B+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {1}{231} (2 (99 A+88 B+80 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (99 A+88 B+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {(4 (99 A+88 B+80 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}{1155 a}\\ &=\frac {2 a (99 A+88 B+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {1}{495} (2 (99 A+88 B+80 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {4 a (99 A+88 B+80 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (99 A+88 B+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 C \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}\\ \end {align*}

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Mathematica [A]
time = 1.85, size = 185, normalized size = 0.77 \begin {gather*} \frac {(1089 A+968 B+1510 C+(2871 A+3322 B+3020 C) \cos (c+d x)+13 (99 A+88 B+80 C) \cos (2 (c+d x))+1287 A \cos (3 (c+d x))+1144 B \cos (3 (c+d x))+1040 C \cos (3 (c+d x))+198 A \cos (4 (c+d x))+176 B \cos (4 (c+d x))+160 C \cos (4 (c+d x))+198 A \cos (5 (c+d x))+176 B \cos (5 (c+d x))+160 C \cos (5 (c+d x))) \sec ^5(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{3465 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((1089*A + 968*B + 1510*C + (2871*A + 3322*B + 3020*C)*Cos[c + d*x] + 13*(99*A + 88*B + 80*C)*Cos[2*(c + d*x)]
+ 1287*A*Cos[3*(c + d*x)] + 1144*B*Cos[3*(c + d*x)] + 1040*C*Cos[3*(c + d*x)] + 198*A*Cos[4*(c + d*x)] + 176*
B*Cos[4*(c + d*x)] + 160*C*Cos[4*(c + d*x)] + 198*A*Cos[5*(c + d*x)] + 176*B*Cos[5*(c + d*x)] + 160*C*Cos[5*(c
+ d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3465*d)

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Maple [A]
time = 14.72, size = 204, normalized size = 0.85

method	result	size
default	\(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (1584 A \left (\cos ^{5}\left (d x +c \right )\right )+1408 B \left (\cos ^{5}\left (d x +c \right )\right )+1280 C \left (\cos ^{5}\left (d x +c \right )\right )+792 A \left (\cos ^{4}\left (d x +c \right )\right )+704 B \left (\cos ^{4}\left (d x +c \right )\right )+640 C \left (\cos ^{4}\left (d x +c \right )\right )+594 A \left (\cos ^{3}\left (d x +c \right )\right )+528 B \left (\cos ^{3}\left (d x +c \right )\right )+480 C \left (\cos ^{3}\left (d x +c \right )\right )+495 A \left (\cos ^{2}\left (d x +c \right )\right )+440 B \left (\cos ^{2}\left (d x +c \right )\right )+400 C \left (\cos ^{2}\left (d x +c \right )\right )+385 B \cos \left (d x +c \right )+350 C \cos \left (d x +c \right )+315 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3465 d \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}\)	\(204\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465/d*(-1+cos(d*x+c))*(1584*A*cos(d*x+c)^5+1408*B*cos(d*x+c)^5+1280*C*cos(d*x+c)^5+792*A*cos(d*x+c)^4+704*
B*cos(d*x+c)^4+640*C*cos(d*x+c)^4+594*A*cos(d*x+c)^3+528*B*cos(d*x+c)^3+480*C*cos(d*x+c)^3+495*A*cos(d*x+c)^2+
440*B*cos(d*x+c)^2+400*C*cos(d*x+c)^2+385*B*cos(d*x+c)+350*C*cos(d*x+c)+315*C)*(a*(1+cos(d*x+c))/cos(d*x+c))^(
1/2)/cos(d*x+c)^5/sin(d*x+c)

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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)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

16/3465*(3465*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*((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(14*d*x + 14
*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos
(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) +
cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d
*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*
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(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x
+ 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*si
n(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x +
10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*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(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c
) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*
d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*
sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*
x + 6*c)*sin(2*d*x + 2*c) - 6*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(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)
*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x
+ 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(
2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x +
6*c)*sin(2*d*x + 2*c) + 6*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(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c)
+ 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*
cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6
*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x +
8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*
c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c
)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*
c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 1
0*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) +
sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x +
4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x
+ 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*
d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*
x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*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*
(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*
c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2
+ 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) +
225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c)
+ 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^
2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c
) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c)...

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Fricas [A]
time = 2.99, size = 151, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (16 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{5} + 8 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right ) + 315 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/3465*(16*(99*A + 88*B + 80*C)*cos(d*x + c)^5 + 8*(99*A + 88*B + 80*C)*cos(d*x + c)^4 + 6*(99*A + 88*B + 80*C
)*cos(d*x + c)^3 + 5*(99*A + 88*B + 80*C)*cos(d*x + c)^2 + 35*(11*B + 10*C)*cos(d*x + c) + 315*C)*sqrt((a*cos(
d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)

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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 + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(A+B*sec(d*x+c)+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 + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**4, x)

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Giac [A]
time = 1.19, size = 410, normalized size = 1.72 \begin {gather*} -\frac {2 \, {\left (3465 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (10395 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 8085 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5775 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (15246 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 14322 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 16170 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (14058 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13266 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 8910 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (6633 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4741 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5885 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (891 \, \sqrt {2} A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1177 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 755 \, \sqrt {2} 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}\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 )}{3465 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} \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)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-2/3465*(3465*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 3465*sqrt(2)*B*a^6*sgn(cos(d*x + c)) + 3465*sqrt(2)*C*a^6*sgn(
cos(d*x + c)) - (10395*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 8085*sqrt(2)*B*a^6*sgn(cos(d*x + c)) + 5775*sqrt(2)*C
*a^6*sgn(cos(d*x + c)) - (15246*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 14322*sqrt(2)*B*a^6*sgn(cos(d*x + c)) + 1617
0*sqrt(2)*C*a^6*sgn(cos(d*x + c)) - (14058*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 13266*sqrt(2)*B*a^6*sgn(cos(d*x +
c)) + 8910*sqrt(2)*C*a^6*sgn(cos(d*x + c)) - (6633*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 4741*sqrt(2)*B*a^6*sgn(c
os(d*x + c)) + 5885*sqrt(2)*C*a^6*sgn(cos(d*x + c)) - (891*sqrt(2)*A*a^6*sgn(cos(d*x + c)) + 1177*sqrt(2)*B*a^
6*sgn(cos(d*x + c)) + 755*sqrt(2)*C*a^6*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)/((a*tan(1/2*d*x + 1/
2*c)^2 - a)^5*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*d)

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Mupad [B]
time = 12.22, size = 724, normalized size = 3.03 \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\,16{}\mathrm {i}}{11\,d}-\frac {\left (32\,A+32\,B+64\,C\right )\,1{}\mathrm {i}}{11\,d}+\frac {\left (16\,A+32\,B\right )\,1{}\mathrm {i}}{11\,d}\right )+\frac {A\,16{}\mathrm {i}}{11\,d}-\frac {\left (32\,A+32\,B+64\,C\right )\,1{}\mathrm {i}}{11\,d}+\frac {\left (16\,A+32\,B\right )\,1{}\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 (\frac {C\,64{}\mathrm {i}}{9\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (-\frac {A\,16{}\mathrm {i}}{9\,d}+\frac {C\,256{}\mathrm {i}}{33\,d}+\frac {\left (176\,A+352\,B+704\,C\right )\,1{}\mathrm {i}}{99\,d}\right )-\frac {\left (176\,A+352\,B\right )\,1{}\mathrm {i}}{99\,d}+\frac {\left (176\,A+704\,C\right )\,1{}\mathrm {i}}{99\,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\,16{}\mathrm {i}}{7\,d}+\frac {\left (352\,B+896\,C\right )\,1{}\mathrm {i}}{693\,d}+\frac {\left (3168\,B+6336\,C\right )\,1{}\mathrm {i}}{693\,d}\right )+\frac {\left (1584\,A+3168\,B\right )\,1{}\mathrm {i}}{693\,d}+\frac {\left (3168\,B-6336\,C\right )\,1{}\mathrm {i}}{693\,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 ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,16{}\mathrm {i}}{5\,d}-\frac {\left (352\,B-528\,A+320\,C\right )\,1{}\mathrm {i}}{1155\,d}\right )+\frac {\left (3696\,A+7392\,B\right )\,1{}\mathrm {i}}{1155\,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 )}^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 (3168\,A+2816\,B+2560\,C\right )\,1{}\mathrm {i}}{3465\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )}-\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 (1584\,A+1408\,B+1280\,C\right )\,1{}\mathrm {i}}{3465\,d\,\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^4,x)

[Out]

((exp(c*1i + d*x*1i)*((A*16i)/(5*d) - ((352*B - 528*A + 320*C)*1i)/(1155*d)) + ((3696*A + 7392*B)*1i)/(1155*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)^2) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((C*64i)/(9*d) + exp(c*1i + d*x*1i)*
((C*256i)/(33*d) - (A*16i)/(9*d) + ((176*A + 352*B + 704*C)*1i)/(99*d)) - ((176*A + 352*B)*1i)/(99*d) + ((176*
A + 704*C)*1i)/(99*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)*(((1584*A + 3168*B)*1i)/(693*d) - exp(c*1i + d*x*1i)*(((352*B + 896*C)*1i)/(6
93*d) - (A*16i)/(7*d) + ((3168*B + 6336*C)*1i)/(693*d)) + ((3168*B - 6336*C)*1i)/(693*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)/(11*d) - ((32*A + 32*B + 64*C)*1i)/(11*d) + ((16*A + 32*B)*1i)/(11*d)) + (A*16i)/(11*d) - (
(32*A + 32*B + 64*C)*1i)/(11*d) + ((16*A + 32*B)*1i)/(11*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) +
1)^5) - (exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(3168*A + 2816*B + 2
560*C)*1i)/(3465*d*(exp(c*1i + d*x*1i) + 1)) - (exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i +
d*x*1i)/2))^(1/2)*(1584*A + 1408*B + 1280*C)*1i)/(3465*d*(exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1))

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