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

Optimal. Leaf size=307 \[ \frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d}+\frac{4 a^3 (27 A+21 B+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d} \]

[Out]

(-4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*
(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A +
21*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(42*A + 41*B + 32*C)*Sec[c + d*x]^(3/2)*Sin[c +
d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sec[c + d*
x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sec[c + d*x]^(3/2)*(a^3 +
a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)

________________________________________________________________________________________

Rubi [A]  time = 0.645568, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4088, 4018, 3997, 3787, 3771, 2641, 3768, 2639} \[ \frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d}+\frac{4 a^3 (27 A+21 B+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*
(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A +
21*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(42*A + 41*B + 32*C)*Sec[c + d*x]^(3/2)*Sin[c +
d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sec[c + d*
x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sec[c + d*x]^(3/2)*(a^3 +
a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)

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]

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \sqrt{\sec (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{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \left (\frac{1}{2} a (9 A+C)+\frac{3}{2} a (3 B+2 C) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{4 \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac{1}{4} a^2 (63 A+9 B+13 C)+\frac{1}{4} a^2 (63 A+99 B+73 C) \sec (c+d x)\right ) \, dx}{63 a}\\ &=\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac{8 \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac{3}{4} a^3 (63 A+24 B+23 C)+\frac{9}{4} a^3 (42 A+41 B+32 C) \sec (c+d x)\right ) \, dx}{315 a}\\ &=\frac{4 a^3 (42 A+41 B+32 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac{16 \int \sqrt{\sec (c+d x)} \left (\frac{45}{8} a^4 (21 A+13 B+11 C)+\frac{63}{8} a^4 (27 A+21 B+17 C) \sec (c+d x)\right ) \, dx}{945 a}\\ &=\frac{4 a^3 (42 A+41 B+32 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac{1}{21} \left (2 a^3 (21 A+13 B+11 C)\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (2 a^3 (27 A+21 B+17 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}-\frac{1}{15} \left (2 a^3 (27 A+21 B+17 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^3 (21 A+13 B+11 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+13 B+11 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}-\frac{1}{15} \left (2 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^3 (27 A+21 B+17 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 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2 (3 B+2 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac{2 (63 A+99 B+73 C) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}\\ \end{align*}

Mathematica [C]  time = 7.19545, size = 1267, normalized size = 4.13 \[ \frac{3 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 ) \text{Hypergeometric2F1}\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 )}{5 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{7 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 ) \text{Hypergeometric2F1}\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 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 ) \text{Hypergeometric2F1}\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{A \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),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 )}{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{13 B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),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 C \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),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{(\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 ^4(c+d x)}{18 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+27 C \sin (d x)) \sec ^3(c+d x)}{126 d}+\frac{\sec (c) (45 B \sin (c)+135 C \sin (c)+63 A \sin (d x)+189 B \sin (d x)+238 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (63 A \sin (c)+189 B \sin (c)+238 C \sin (c)+315 A \sin (d x)+390 B \sin (d x)+330 C \sin (d x)) \sec (c+d x)}{630 d}+\frac{(27 A+21 B+17 C) \cos (d x) \csc (c)}{15 d}+\frac{(21 A+26 B+22 C) \tan (c)}{42 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[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(3*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))/(5*Sqrt[2]*d*E^
(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*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))/(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*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))/(
45*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (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))/(d*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (13*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)) + (11*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*S
ec[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)) + (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)*(((27*A + 21*B + 17*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])
/(18*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 9*B*Sin[d*x] + 27*C*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^2
*(45*B*Sin[c] + 135*C*Sin[c] + 63*A*Sin[d*x] + 189*B*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x
]*(63*A*Sin[c] + 189*B*Sin[c] + 238*C*Sin[c] + 315*A*Sin[d*x] + 390*B*Sin[d*x] + 330*C*Sin[d*x]))/(630*d) + ((
21*A + 26*B + 22*C)*Tan[c])/(42*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))

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Maple [B]  time = 11.043, size = 1265, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),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)*(2*A*(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)*(-1/56*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)/(cos(1/2*d*x
+1/2*c)^2-1/2)^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)/(cos(1/2*d*x+1/2
*c)^2-1/2)^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)))-16/5*(1/8*A+3/8*B+3/8*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*sin(1/2*d*x+1/2*c)^
6*cos(1/2*d*x+1/2*c)-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*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)-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)+16*(3/8*A+3/8*B+1/8*C)*(-1/6*cos(1/2*d*x+1/2*c)*(-2*si
n(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)^2-1/2)^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)*EllipticF(cos(1/2*d*x+1/
2*c),2^(1/2)))+2*C*(-1/144*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)/(cos(1/2*d*
x+1/2*c)^2-1/2)^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)/(cos(1/2*d*x+1
/2*c)^2-1/2)^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)*EllipticF(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))-El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+16*(3/8*A+1/8*B)*(-(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*sin(1/2*d*x+1/2*c)^4+sin(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))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{3} \sec \left (d x + c\right )^{5} +{\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} +{\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} +{\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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