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

Optimal. Leaf size=277 \[ \frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 (9 A+13 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d}+\frac{4 a^3 (21 A+17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]

[Out]

(-4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A +
11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(21*A + 17*B)*Sqrt[Sec
[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(13*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^3*(24*A
+ 23*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*
x])/(9*d) + (2*(9*A + 13*B)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d)

________________________________________________________________________________________

Rubi [A]  time = 0.540878, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac{4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 (9 A+13 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d}+\frac{4 a^3 (21 A+17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]

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]),x]

[Out]

(-4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A +
11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(21*A + 17*B)*Sqrt[Sec
[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(13*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^3*(24*A
+ 23*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*
x])/(9*d) + (2*(9*A + 13*B)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d)

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 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 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]

Rubi steps

\begin{align*} \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac{3}{2} a (3 A+B)+\frac{1}{2} a (9 A+13 B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{4}{63} \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac{15}{2} a^2 (3 A+2 B)+\frac{3}{2} a^2 (24 A+23 B) \sec (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (24 A+23 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{8}{315} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{21}{4} a^3 (21 A+17 B)+\frac{45}{4} a^3 (13 A+11 B) \sec (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (24 A+23 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{4 a^3 (21 A+17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (13 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{4 a^3 (24 A+23 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a^3 (21 A+17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (13 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{4 a^3 (24 A+23 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{21} \left (2 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{15} \left (2 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^3 (21 A+17 B) \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 (13 A+11 B) \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 (21 A+17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (13 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{4 a^3 (24 A+23 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end{align*}

Mathematica [C]  time = 6.80617, size = 793, normalized size = 2.86 \[ \frac{7 A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{30 \sqrt{2} d (A \cos (c+d x)+B)}+\frac{17 B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{90 \sqrt{2} d (A \cos (c+d x)+B)}+\frac{13 A \sqrt{\cos (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac{11 B \sqrt{\cos (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac{\sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x)) \left (\frac{(21 A+17 B) \csc (c) \cos (d x)}{30 d}+\frac{\sec (c) \sec ^3(c+d x) (9 A \sin (d x)+7 B \sin (c)+27 B \sin (d x))}{252 d}+\frac{\sec (c) \sec ^2(c+d x) (45 A \sin (c)+189 A \sin (d x)+135 B \sin (c)+238 B \sin (d x))}{1260 d}+\frac{\sec (c) \sec (c+d x) (189 A \sin (c)+390 A \sin (d x)+238 B \sin (c)+330 B \sin (d x))}{1260 d}+\frac{(13 A+11 B) \tan (c)}{42 d}+\frac{B \sec (c) \sin (d x) \sec ^4(c+d x)}{36 d}\right )}{\sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)} \]

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]),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]^4*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]))/(30*Sqrt[2]*d*E^(I*d*x)*(B + A*Cos
[c + 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]
^4*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]))/(90*Sqrt[2]*d*E^(I*
d*x)*(B + A*Cos[c + d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Se
c[c + d*x])^3*(A + B*Sec[c + d*x]))/(42*d*(B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + (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]))/(42*d*(B + A*Cos[c
+ d*x])*Sec[c + d*x]^(7/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(((21*A + 17*
B)*Cos[d*x]*Csc[c])/(30*d) + (B*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^3*(7*B*Sin[c] +
9*A*Sin[d*x] + 27*B*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^2*(45*A*Sin[c] + 135*B*Sin[c] + 189*A*Sin[d*x] +
238*B*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(189*A*Sin[c] + 238*B*Sin[c] + 390*A*Sin[d*x] + 330*B*Sin[d*
x]))/(1260*d) + ((13*A + 11*B)*Tan[c])/(42*d)))/((B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2))

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Maple [B]  time = 8.211, size = 1180, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

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)),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*(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)/(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)))-16/5*(3/8*A+3/8*B)/(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*(1/8*A+3/8*B)*(-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+si
n(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*B*(-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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+2*A*(-(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+s
in(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(sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),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 (B a^{3} \sec \left (d x + c\right )^{5} +{\left (A + 3 \, B\right )} a^{3} \sec \left (d x + c\right )^{4} + 3 \,{\left (A + B\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 ) \end{align*}

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)),x, algorithm="fricas")

[Out]

integral((B*a^3*sec(d*x + c)^5 + (A + 3*B)*a^3*sec(d*x + c)^4 + 3*(A + B)*a^3*sec(d*x + c)^3 + (3*A + B)*a^3*s
ec(d*x + c)^2 + A*a^3*sec(d*x + c))*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(sec(d*x+c)**(3/2)*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (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} \end{align*}

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)),x, algorithm="giac")

[Out]

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