3.549 \(\int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=343 \[ \frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{693 d}+\frac {4 a^3 (21 A+17 B+15 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (11 B+6 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{99 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]

[Out]

4/231*a^3*(143*A+121*B+105*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+4/1155*a^3*(264*A+253*B+210*C)*sec(d*x+c)^(5/2)*si
n(d*x+c)/d+2/11*C*sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+2/99*(11*B+6*C)*sec(d*x+c)^(5/2)*(a^2+a^2*s
ec(d*x+c))^2*sin(d*x+c)/a/d+2/693*(99*A+143*B+105*C)*sec(d*x+c)^(5/2)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d+4/15*a
^3*(21*A+17*B+15*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/d-4/15*a^3*(21*A+17*B+15*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1
/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/231*a^3*(143*A+121*B
+105*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)
*sec(d*x+c)^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.70, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4088, 4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{693 d}+\frac {4 a^3 (21 A+17 B+15 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (11 B+6 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{99 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*
(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(21*
A + 17*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sec[c + d*x]^(3/2)*S
in[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*C*Sec[c +
d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(11*B + 6*C)*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c
+ d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(99*A + 143*B + 105*C)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c
+ d*x])/(693*d)

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f
, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rule 4018

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] && Ne
Q[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1]

Rule 4088

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 ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (\frac {1}{2} a (11 A+3 C)+\frac {1}{2} a (11 B+6 C) \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {4 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {3}{4} a^2 (33 A+11 B+15 C)+\frac {1}{4} a^2 (99 A+143 B+105 C) \sec (c+d x)\right ) \, dx}{99 a}\\ &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {8 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {15}{4} a^3 (33 A+22 B+21 C)+\frac {3}{4} a^3 (264 A+253 B+210 C) \sec (c+d x)\right ) \, dx}{693 a}\\ &=\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {16 \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {231}{8} a^4 (21 A+17 B+15 C)+\frac {45}{8} a^4 (143 A+121 B+105 C) \sec (c+d x)\right ) \, dx}{3465 a}\\ &=\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{77} \left (2 a^3 (143 A+121 B+105 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}-\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (2 a^3 (143 A+121 B+105 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}-\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 7.41, size = 1324, normalized size = 3.86 \[ \frac {7 A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 \sqrt {2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {17 B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{45 \sqrt {2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 \sqrt {2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {13 A \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {11 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {5 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{11 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {(\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac {C \sec (c) \sin (d x) \sec ^5(c+d x)}{22 d}+\frac {\sec (c) (9 C \sin (c)+11 B \sin (d x)+33 C \sin (d x)) \sec ^4(c+d x)}{198 d}+\frac {\sec (c) (77 B \sin (c)+231 C \sin (c)+99 A \sin (d x)+297 B \sin (d x)+378 C \sin (d x)) \sec ^3(c+d x)}{1386 d}+\frac {\sec (c) (495 A \sin (c)+1485 B \sin (c)+1890 C \sin (c)+2079 A \sin (d x)+2618 B \sin (d x)+2310 C \sin (d x)) \sec ^2(c+d x)}{6930 d}+\frac {\sec (c) (2079 A \sin (c)+2618 B \sin (c)+2310 C \sin (c)+4290 A \sin (d x)+3630 B \sin (d x)+3150 C \sin (d x)) \sec (c+d x)}{6930 d}+\frac {(21 A+17 B+15 C) \cos (d x) \csc (c)}{15 d}+\frac {(143 A+121 B+105 C) \tan (c)}{231 d}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*S
qrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c
+ d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E
^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (17*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*
x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-
1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d
*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x])) + (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5
*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4,
-E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3
*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(
c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A +
2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (11*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*
x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C +
2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (5*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2
]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(11*d*(A + 2*C + 2*B*Co
s[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec
[c + d*x] + C*Sec[c + d*x]^2)*(((21*A + 17*B + 15*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*
x])/(22*d) + (Sec[c]*Sec[c + d*x]^4*(9*C*Sin[c] + 11*B*Sin[d*x] + 33*C*Sin[d*x]))/(198*d) + (Sec[c]*Sec[c + d*
x]^3*(77*B*Sin[c] + 231*C*Sin[c] + 99*A*Sin[d*x] + 297*B*Sin[d*x] + 378*C*Sin[d*x]))/(1386*d) + (Sec[c]*Sec[c
+ d*x]^2*(495*A*Sin[c] + 1485*B*Sin[c] + 1890*C*Sin[c] + 2079*A*Sin[d*x] + 2618*B*Sin[d*x] + 2310*C*Sin[d*x]))
/(6930*d) + (Sec[c]*Sec[c + d*x]*(2079*A*Sin[c] + 2618*B*Sin[c] + 2310*C*Sin[c] + 4290*A*Sin[d*x] + 3630*B*Sin
[d*x] + 3150*C*Sin[d*x]))/(6930*d) + ((143*A + 121*B + 105*C)*Tan[c])/(231*d)))/((A + 2*C + 2*B*Cos[c + d*x] +
A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))

________________________________________________________________________________________

fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \sec \left (d x + c\right )^{6} + {\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{5} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right )^{2} + A a^{3} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*a^3*sec(d*x + c)^6 + (B + 3*C)*a^3*sec(d*x + c)^5 + (A + 3*B + 3*C)*a^3*sec(d*x + c)^4 + (3*A + 3*
B + C)*a^3*sec(d*x + c)^3 + (3*A + B)*a^3*sec(d*x + c)^2 + A*a^3*sec(d*x + c))*sqrt(sec(d*x + c)), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 24.13, size = 1427, normalized size = 4.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-16/5*(3/8*A+3/8*B+1/8*C)/(8*sin(1/2*d*x+1/2*c
)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^
(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c
)*sin(1/2*d*x+1/2*c)^6-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+3*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2
*c))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*A*(-(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)
^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(
-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c
)^2/(2*sin(1/2*d*x+1/2*c)^2-1)+2*C*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(
1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/
2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2
)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1
/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+16*(1/8*B+3/8*C)*(-1/144*co
s(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^5-7/180*cos(
1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^3-14/15*sin(1/
2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti
cF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2
*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),
2^(1/2))))+16*(1/8*A+3/8*B+3/8*C)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/
2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/
(-1/2+cos(1/2*d*x+1/2*c)^2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*
d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+16*(3/8*A+1/8*B)*(-1/6*cos(1/2
*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+1/3*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellip
ticF(cos(1/2*d*x+1/2*c),2^(1/2))))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

int((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________