3.1316 \(\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=384 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{5 d}+\frac{2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 (-(35 A-87 C))+98 a b B+5 b^2 (7 A+5 C)\right )}{105 d}+\frac{2 b \sin (c+d x) \left (413 a^2 b B+192 a^3 C+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
[Out]
(2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d
) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x
)/2, 2])/(21*d) + (2*b*(413*a^2*b*B + 63*b^3*B + 192*a^3*C + 2*a*b^2*(175*A + 101*C))*Sin[c + d*x])/(105*d*Sqr
t[Cos[c + d*x]]) - (2*a^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/
(105*d) + (2*(35*A*b^2 + 77*a*b*B + 48*a^2*C + 25*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d
*x]^(3/2)) + (2*(7*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*C
os[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))
________________________________________________________________________________________
Rubi [A] time = 1.31648, antiderivative size = 384, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used =
{4112, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{5 d}+\frac{2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 (-(35 A-87 C))+98 a b B+5 b^2 (7 A+5 C)\right )}{105 d}+\frac{2 b \sin (c+d x) \left (413 a^2 b B+192 a^3 C+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d
) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x
)/2, 2])/(21*d) + (2*b*(413*a^2*b*B + 63*b^3*B + 192*a^3*C + 2*a*b^2*(175*A + 101*C))*Sin[c + d*x])/(105*d*Sqr
t[Cos[c + d*x]]) - (2*a^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/
(105*d) + (2*(35*A*b^2 + 77*a*b*B + 48*a^2*C + 25*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d
*x]^(3/2)) + (2*(7*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*C
os[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))
Rule 4112
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] && !IntegerQ[n] && IntegerQ[m]
Rule 3047
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Rule 3031
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]
Rule 3023
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, 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 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 \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \frac{(b+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{(b+a \cos (c+d x))^3 \left (\frac{1}{2} (7 b B+8 a C)+\frac{1}{2} (7 A b+7 a B+5 b C) \cos (c+d x)+\frac{1}{2} a (7 A-3 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4}{35} \int \frac{(b+a \cos (c+d x))^2 \left (\frac{1}{4} \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right )+\frac{1}{4} \left (70 a A b+35 a^2 B+21 b^2 B+34 a b C\right ) \cos (c+d x)+\frac{1}{4} a (35 a A-21 b B-39 a C) \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8}{105} \int \frac{(b+a \cos (c+d x)) \left (\frac{1}{8} \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right )+\frac{1}{8} \left (105 a^3 B+77 a b^2 B+5 b^3 (7 A+5 C)+a^2 (315 A b+33 b C)\right ) \cos (c+d x)-\frac{3}{8} a \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right ) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{16}{105} \int \frac{\frac{1}{16} \left (-518 a^3 b B-140 a b^3 B-192 a^4 C-5 b^4 (7 A+5 C)-5 a^2 b^2 (133 A+47 C)\right )-\frac{21}{16} \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \cos (c+d x)+\frac{3}{16} a^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right ) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}-\frac{2 a^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{32}{315} \int \frac{-\frac{15}{32} \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right )-\frac{63}{32} \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right ) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}-\frac{2 a^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{5} \left (-5 a^4 B+30 a^2 b^2 B+3 b^4 B-20 a^3 b (A-C)+4 a b^3 (5 A+3 C)\right ) \int \sqrt{\cos (c+d x)} \, dx-\frac{1}{21} \left (-84 a^3 b B-28 a b^3 B-42 a^2 b^2 (3 A+C)-7 a^4 (A+3 C)-b^4 (7 A+5 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right ) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}-\frac{2 a^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 9.44671, size = 4791, normalized size = 12.48 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
((4*I)*a^3*A*b*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*
I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*C
os[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*
c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2,
3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*S
in[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos
[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x
])) - ((4*I)*a*A*b^3*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*
E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d
*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*
Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4,
1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d
*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x
))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c +
2*d*x])) + (I*a^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*
E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d
*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*
Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4,
1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d
*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x
))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c +
2*d*x])) - ((6*I)*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]
^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^
((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*
I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2
F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^
((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((
2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*C
os[2*c + 2*d*x])) - (((3*I)/5)*b^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[
c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2
*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] +
I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hyperge
ometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*
(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(
1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])) - ((4*I)*a^3*b*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*S
qrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2
*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*H
ypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] +
(2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-
I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[
c + d*x] + A*Cos[2*c + 2*d*x])) - (((12*I)/5)*a*b^3*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[
c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Si
n[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)
*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin
[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x)
)*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin
[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2)*((-2*(20*a^3*A*b - 40*a*A*b^3 + 5*a^4*B - 60*a^2*b^2*B - 6*b^4*B - 40*a^3*b*C - 24*a*b^3*C
+ 20*a^3*A*b*Cos[2*c] + 5*a^4*B*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a^4*A*Cos[d*x]*Sin[c])/(3*d) + (4*a^4*A*Co
s[c]*Sin[d*x])/(3*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (4*Sec[c]*Sec[c + d*x]^3*(5*b^4*C*Sin[
c] + 7*b^4*B*Sin[d*x] + 28*a*b^3*C*Sin[d*x]))/(35*d) + (4*Sec[c]*Sec[c + d*x]*(35*A*b^4*Sin[c] + 140*a*b^3*B*S
in[c] + 210*a^2*b^2*C*Sin[c] + 25*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2*B*Sin[d*x] + 63*b^4*B*Sin[
d*x] + 420*a^3*b*C*Sin[d*x] + 252*a*b^3*C*Sin[d*x]))/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(21*b^4*B*Sin[c] + 84*
a*b^3*C*Sin[c] + 35*A*b^4*Sin[d*x] + 140*a*b^3*B*Sin[d*x] + 210*a^2*b^2*C*Sin[d*x] + 25*b^4*C*Sin[d*x]))/(105*
d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^4*A*Cos[c + d*x]^6*Csc[
c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x
] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*
Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C +
2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (24*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*Hypergeometr
icPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d
*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x
- ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x]
+ A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^4*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4
}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTa
n[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sq
rt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x
])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcT
an[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1
- Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x -
ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^
2]) - (16*a*b^3*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a +
b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan
[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(
3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^4*C*Co
s[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(
A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sq
rt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x
])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*b^2*C*Cos[c + d*x]^6*Csc[c
]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]
+ C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*S
in[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*
B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{
1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*
Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTa
n[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A
*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])
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Maple [B] time = 12.151, size = 1624, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4/3*A*a^4*(2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*
c)+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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*(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)-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)+(-4*A*a^4+8*A*a^3*b+2*B*a^4)*
(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*a^4*(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(co
s(1/2*d*x+1/2*c),2^(1/2))-8*A*a^3*b*(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))+12*A*a^2*b^2*(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(c
os(1/2*d*x+1/2*c),2^(1/2))-2*B*a^4*(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))+8*B*a^3*b*(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*a^4*C*(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*b^2*(A*b^2+4*B*a*b+6*C*a^2)*(-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)))+2*C*b^4*(-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)))-2/5*b^3*(B*b+4*C*
a)/(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*(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)*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*(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)*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*(2*si
n(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)-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)+4*a*b*(2*A*b^2+3*B*a*b+2*C*a^
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)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)
^2)^(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))/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(cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^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 b^{4} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{5} + A a^{4} \cos \left (d x + c\right ) +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*b^4*cos(d*x + c)*sec(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*cos(d*x + c)*sec(d*x + c)^5 + A*a^4*cos(d*x
+ c) + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4)*cos(d*x + c)*sec(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)
*cos(d*x + c)*sec(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*cos(d*x + c)*sec(d*x + c)^2 + (B*a^4 + 4*A*a^
3*b)*cos(d*x + c)*sec(d*x + c))*sqrt(cos(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(cos(d*x+c)**(3/2)*(a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**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 (b \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*(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)*(b*sec(d*x + c) + a)^4*cos(d*x + c)^(3/2), x)