\(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\) [363]
Optimal result
Integrand size = 33, antiderivative size = 566 \[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (6 a b^3 (209 A-505 B)-3 b^4 (539 A-225 B)-15 a^2 b^2 (121 A-19 B)+40 a^4 B-a^3 (110 A b-30 b B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^3 d}-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}
\]
[Out]
2/3465*(a-b)*(110*A*a^4*b-3069*A*a^2*b^3-1617*A*b^5-40*B*a^5-255*B*a^3*b^2-3705*B*a*b^4)*cot(d*x+c)*EllipticE(
(a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(
d*x+c))/(a-b))^(1/2)/b^4/d-2/3465*(a-b)*(6*a*b^3*(209*A-505*B)-3*b^4*(539*A-225*B)-15*a^2*b^2*(121*A-19*B)+40*
B*a^4-a^3*(110*A*b-30*B*b))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)
^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^3/d-2/3465*(110*A*a^2*b-539*A*b^3-40*B
*a^3-335*B*a*b^2)*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/b^2/d-2/693*(22*A*a*b-8*B*a^2-81*B*b^2)*(a+b*sec(d*x+c))^(
5/2)*tan(d*x+c)/b^2/d+2/99*(11*A*b-4*B*a)*(a+b*sec(d*x+c))^(7/2)*tan(d*x+c)/b^2/d+2/11*B*sec(d*x+c)*(a+b*sec(d
*x+c))^(7/2)*tan(d*x+c)/b/d-2/3465*(110*A*a^3*b-1254*A*a*b^3-40*B*a^4-285*B*a^2*b^2-675*B*b^4)*(a+b*sec(d*x+c)
)^(1/2)*tan(d*x+c)/b^2/d
Rubi [A] (verified)
Time = 1.93 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4118, 4167, 4087, 4090, 3917,
4089} \[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=-\frac {2 \left (-8 a^2 B+22 a A b-81 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \left (-40 a^3 B+110 a^2 A b-335 a b^2 B-539 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (40 a^4 B-a^3 (110 A b-30 b B)-15 a^2 b^2 (121 A-19 B)+6 a b^3 (209 A-505 B)-3 b^4 (539 A-225 B)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{3465 b^3 d}-\frac {2 \left (-40 a^4 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 b^4 B\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3465 b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (-40 a^5 B+110 a^4 A b-255 a^3 b^2 B-3069 a^2 A b^3-3705 a b^4 B-1617 A b^5\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3465 b^4 d}+\frac {2 (11 A b-4 a B) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac {2 B \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}
\]
[In]
Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]
[Out]
(2*(a - b)*Sqrt[a + b]*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*C
ot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]
))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(a - b)*Sqrt[a + b]*(6*a*b^3*(209*A - 5
05*B) - 3*b^4*(539*A - 225*B) - 15*a^2*b^2*(121*A - 19*B) + 40*a^4*B - a^3*(110*A*b - 30*b*B))*Cot[c + d*x]*El
lipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sq
rt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*
B - 675*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*b^3 - 40*a^3*B -
335*a*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B - 81*b^2*B)*(a + b
*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/
(99*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)
Rule 3917
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4087
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[1/(m + 1), Int[Csc[e + f
*x]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Rule 4089
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Rule 4090
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[Csc[e + f*x]*((1 +
Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]
Rule 4118
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-B)*d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/
(b*f*(m + n))), x] + Dist[d^2/(b*(m + n)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*Simp[a*B*(n - 2
) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e
, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && !IGtQ[m, 1]
Rule 4167
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*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*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*
B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {2 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (a B+\frac {9}{2} b B \sec (c+d x)+\frac {1}{2} (11 A b-4 a B) \sec ^2(c+d x)\right ) \, dx}{11 b} \\ & = \frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {1}{4} b (77 A b-10 a B)-\frac {1}{4} \left (22 a A b-8 a^2 B-81 b^2 B\right ) \sec (c+d x)\right ) \, dx}{99 b^2} \\ & = -\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {8 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {3}{8} b \left (143 a A b-10 a^2 B+135 b^2 B\right )-\frac {1}{8} \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) \sec (c+d x)\right ) \, dx}{693 b^2} \\ & = -\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {16 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{16} b \left (605 a^2 A b+539 A b^3-10 a^3 B+1010 a b^2 B\right )-\frac {3}{16} \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sec (c+d x)\right ) \, dx}{3465 b^2} \\ & = -\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {32 \int \frac {\sec (c+d x) \left (\frac {3}{32} b \left (1705 a^3 A b+2871 a A b^3+10 a^4 B+3315 a^2 b^2 B+675 b^4 B\right )-\frac {3}{32} \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{10395 b^2} \\ & = -\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}-\frac {\left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 b^2}-\frac {\left ((a-b) \left (6 a b^3 (209 A-505 B)-3 b^4 (539 A-225 B)-15 a^2 b^2 (121 A-19 B)+40 a^4 B-a^3 (110 A b-30 b B)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 b^2} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (6 a b^3 (209 A-505 B)-3 b^4 (539 A-225 B)-15 a^2 b^2 (121 A-19 B)+40 a^4 B-a^3 (110 A b-30 b B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^3 d}-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 B \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(4227\) vs. \(2(566)=1132\).
Time = 26.40 (sec) , antiderivative size = 4227, normalized size of antiderivative = 7.47
\[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Result too large to show}
\]
[In]
Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]
[Out]
(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^
3*b^2*B + 3705*a*b^4*B)*Sin[c + d*x])/(3465*b^3) + (2*Sec[c + d*x]^4*(11*A*b^2*Sin[c + d*x] + 23*a*b*B*Sin[c +
d*x]))/99 + (2*Sec[c + d*x]^3*(209*a*A*b*Sin[c + d*x] + 113*a^2*B*Sin[c + d*x] + 81*b^2*B*Sin[c + d*x]))/693
+ (2*Sec[c + d*x]^2*(825*a^2*A*b*Sin[c + d*x] + 539*A*b^3*Sin[c + d*x] + 15*a^3*B*Sin[c + d*x] + 1145*a*b^2*B*
Sin[c + d*x]))/(3465*b) + (2*Sec[c + d*x]*(55*a^3*A*b*Sin[c + d*x] + 1793*a*A*b^3*Sin[c + d*x] - 20*a^4*B*Sin[
c + d*x] + 1025*a^2*b^2*B*Sin[c + d*x] + 675*b^4*B*Sin[c + d*x]))/(3465*b^2) + (2*b^2*B*Sec[c + d*x]^4*Tan[c +
d*x])/11))/(d*(b + a*Cos[c + d*x])^2) - (2*((2*a^4*A)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3
1*a^2*A*b)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*A*b^3)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[
c + d*x]]) - (17*a^3*B)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*B)/(693*b^2*Sqrt[b + a*Cos[
c + d*x]]*Sqrt[Sec[c + d*x]]) - (247*a*b^2*B)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (124*a^3*A*S
qrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*A*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*
x]]) + (38*a*A*b^2*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (8*a^6*B*Sqrt[Sec[c + d*x]])/(693*b^3*
Sqrt[b + a*Cos[c + d*x]]) - (7*a^4*B*Sqrt[Sec[c + d*x]])/(99*b*Sqrt[b + a*Cos[c + d*x]]) - (26*a^2*b*B*Sqrt[Se
c[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (15*b^3*B*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (3
1*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*A*Cos[2*(c + d*x)]*Sqrt[Se
c[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (7*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b +
a*Cos[c + d*x]]) - (8*a^6*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^4*
B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*b*Sqrt[b + a*Cos[c + d*x]]) - (247*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[S
ec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)
*(2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c
+ d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d
*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*
(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a
+ b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b
^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^
2*Tan[(c + d*x)/2]))/(3465*b^3*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*(-1/3465*(
a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 +
40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a +
b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3
*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*
x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/
2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)
*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[
Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-110*a^4*A*b + 3069
*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[
(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*
(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A
+ 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Ellip
ticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255
*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^3*S
qrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-110*a^4*A*b +
3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[
(c + d*x)/2]^4)/2 + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b
^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a +
b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(
1 + Cos[c + d*x])] - (b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A +
225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Ta
n[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c
+ d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^
5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]],
(a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a +
b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(40*a^4*B - 1
0*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c
+ d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*
(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c
+ d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2
*B + 3705*a*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-110*a^4*A*b + 3069*a^2*A*
b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*
x]*Tan[(c + d*x)/2] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*C
os[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A
+ 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 +
Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c +
d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*
b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])
/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan
[(c + d*x)/2]^2]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-110*a^4*A*b +
3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*S
qrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] -
2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(2
09*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*E
llipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B +
255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(
c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3465*b^3*Sqrt[b +
a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(6883\) vs. \(2(524)=1048\).
Time = 119.07 (sec) , antiderivative size = 6884, normalized size of antiderivative =
12.16
| | |
| method | result | size |
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| parts |
\(\text {Expression too large to display}\) |
\(6884\) |
| default |
\(\text {Expression too large to display}\) |
\(6974\) |
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[In]
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
result too large to display
Fricas [F]
\[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x }
\]
[In]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")
[Out]
integral((B*b^2*sec(d*x + c)^6 + A*a^2*sec(d*x + c)^3 + (2*B*a*b + A*b^2)*sec(d*x + c)^5 + (B*a^2 + 2*A*a*b)*s
ec(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), x)
Sympy [F(-1)]
Timed out. \[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out}
\]
[In]
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)),x)
[Out]
Timed out
Maxima [F(-1)]
Timed out. \[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out}
\]
[In]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")
[Out]
Timed out
Giac [F]
\[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x }
\]
[In]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="giac")
[Out]
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^3, x)
Mupad [F(-1)]
Timed out. \[
\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x
\]
[In]
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^3,x)
[Out]
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^3, x)