3.414 \(\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=345 \[ \frac{2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{231 d}+\frac{2 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a^2 (11 a B+15 A b) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

[Out]

(2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x
]])/(15*d) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]
*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(15*A*b + 11*a*B)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*(9*a^2*
A + 26*A*b^2 + 33*a*b*B)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b
^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sin[c + d
*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.57392, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4025, 4074, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac{2 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a^2 (11 a B+15 A b) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]

[Out]

(2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x
]])/(15*d) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]
*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(15*A*b + 11*a*B)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*(9*a^2*
A + 26*A*b^2 + 33*a*b*B)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b
^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sin[c + d
*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))

Rule 4025

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[a*(a*B*n - A*b*(m - n - 1)) + (
2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; Free
Q[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]

Rule 4074

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_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

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 \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{2}{11} \int \frac{(a+b \sec (c+d x)) \left (-\frac{1}{2} a (15 A b+11 a B)-\frac{1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \sec (c+d x)-\frac{1}{2} b (5 a A+11 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4}{99} \int \frac{\frac{9}{4} a \left (9 a^2 A+26 A b^2+33 a b B\right )+\frac{11}{4} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sec (c+d x)+\frac{9}{4} b^2 (5 a A+11 b B) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4}{99} \int \frac{\frac{9}{4} a \left (9 a^2 A+26 A b^2+33 a b B\right )+\frac{9}{4} b^2 (5 a A+11 b B) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+\frac{1}{9} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{1}{15} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{77} \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{1}{231} \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (\left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \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{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{1}{231} \left (\left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \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{2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \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{2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 3.09557, size = 256, normalized size = 0.74 \[ \frac{\sqrt{\sec (c+d x)} \left (240 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (2 (c+d x)) \left (180 a \left (16 a^2 A+33 a b B+33 A b^2\right ) \cos (2 (c+d x))+154 \left (129 a^2 A b+43 a^3 B+108 a b^2 B+36 A b^3\right ) \cos (c+d x)+15 \left (21 a^3 A \cos (4 (c+d x))+531 a^3 A+1716 a^2 b B+1716 a A b^2+616 b^3 B\right )+770 a^2 (a B+3 A b) \cos (3 (c+d x))\right )+3696 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(3696*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)
/2, 2] + 240*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] +
(154*(129*a^2*A*b + 36*A*b^3 + 43*a^3*B + 108*a*b^2*B)*Cos[c + d*x] + 180*a*(16*a^2*A + 33*A*b^2 + 33*a*b*B)*C
os[2*(c + d*x)] + 770*a^2*(3*A*b + a*B)*Cos[3*(c + d*x)] + 15*(531*a^3*A + 1716*a*A*b^2 + 1716*a^2*b*B + 616*b
^3*B + 21*a^3*A*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(27720*d)

________________________________________________________________________________________

Maple [B]  time = 1.988, size = 825, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*A*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-50400*A*a^3-36960*A*a^2*b-12320*B*a^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a^3+73920*A
*a^2*b+23760*A*a*b^2+24640*B*a^3+23760*B*a^2*b)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-34920*A*a^3-68376*A*
a^2*b-35640*A*a*b^2-5544*A*b^3-22792*B*a^3-35640*B*a^2*b-16632*B*a*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c
)+(13860*A*a^3+31416*A*a^2*b+27720*A*a*b^2+5544*A*b^3+10472*B*a^3+27720*B*a^2*b+16632*B*a*b^2+4620*B*b^3)*sin(
1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2790*A*a^3-5544*A*a^2*b-7920*A*a*b^2-1386*A*b^3-1848*B*a^3-7920*B*a^2*b-
4158*B*a*b^2-2310*B*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+675*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipt
icF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3+2475*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip
ticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a*b^2-4851*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*b-2079*A*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*b^3+2475*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2
)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*b+1155*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(
1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*b^3-1617*B*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3-6237*B*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a*b^2)/(-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*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+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{11}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

integral((B*b^3*sec(d*x + c)^4 + A*a^3 + (3*B*a*b^2 + A*b^3)*sec(d*x + c)^3 + 3*(B*a^2*b + A*a*b^2)*sec(d*x +
c)^2 + (B*a^3 + 3*A*a^2*b)*sec(d*x + c))/sec(d*x + c)^(11/2), 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+b*sec(d*x+c))**3*(A+B*sec(d*x+c))/sec(d*x+c)**(11/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/sec(d*x + c)^(11/2), x)