3.1342 \(\int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=359 \[ \frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{105 a d}+\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A+35 a^2 C+21 a b B-6 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \left (-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (7 a B+3 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{35 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{7 d} \]
[Out]
2/7*A*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/105*(a^2-b^2)*(25*A*a^2-6*A*b^2+21*B*a*b+35*C*a^2
)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*
cos(d*x+c))/(a+b))^(1/2)/a^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+2/35*(3*A*b+7*B*a)*cos(d*x+c)^(3/2)*sin
(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+2/105*(3*A*b^2+42*a*b*B+5*a^2*(5*A+7*C))*sin(d*x+c)*cos(d*x+c)^(1/2)*(a+b*sec
(d*x+c))^(1/2)/a/d-2/105*(6*A*b^3-63*a^3*B-21*a*b^2*B-2*a^2*b*(41*A+70*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/
2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/a^2
/d/((b+a*cos(d*x+c))/(a+b))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 1.38, antiderivative size = 359, normalized size of antiderivative =
1.00, number of steps used = 11, number of rules used = 10, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\)
= 0.222, Rules used = {4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}
\[ \frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{105 a d}+\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A+35 a^2 C+21 a b B-6 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \left (-2 a^2 b (41 A+70 C)-63 a^3 B-21 a b^2 B+6 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (7 a B+3 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{35 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{7 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*
x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*
b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x
]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c +
d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[
c + d*x]]*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)
Rule 2653
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2655
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] && !GtQ[a + b, 0]
Rule 2661
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2663
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] && !GtQ[a + b, 0]
Rule 3856
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 3858
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4035
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(
b_.) + (a_)]), x_Symbol] :> Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(A*b -
a*B)/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && Ne
Q[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Rule 4094
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[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*
Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Rule 4104
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[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Rule 4265
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]
Rubi steps
\begin {align*} \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {1}{2} (3 A b+7 a B)+\frac {1}{2} (5 a A+7 b B+7 a C) \sec (c+d x)+\frac {1}{2} b (2 A+7 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {1}{35} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} \left (44 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \sec (c+d x)+\frac {1}{4} b (16 A b+14 a B+35 b C) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right )-\frac {1}{8} a \left (84 a b B+5 a^2 (5 A+7 C)+3 b^2 (17 A+35 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{105 a}\\ &=\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{105 a^2}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{105 a^2}\\ &=\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{105 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{105 a^2 \sqrt {b+a \cos (c+d x)}}\\ &=\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{105 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{105 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [C] time = 22.86, size = 3261, normalized size = 9.08 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((115*a^2*A + 12*A*b^2
+ 168*a*b*B + 140*a^2*C)*Sin[c + d*x])/(105*a) + (2*(8*A*b + 7*a*B)*Sin[2*(c + d*x)])/35 + (a*A*Sin[3*(c + d*
x)])/7))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((
164*a*A*b*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b^3*Sqrt[Cos[c + d*x]])
/(35*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]
*Sqrt[Sec[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a*b*C
*Sqrt[Cos[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a^2*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (34*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[
c + d*x]]) + (8*a*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*C*Sqrt[Cos[
c + d*x]]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/S
qrt[b + a*Cos[c + d*x]])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x
] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*Arc
Sinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a
+ b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Ta
n[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]
- (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*T
an[(c + d*x)/2]))/(105*a^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c +
d*x]^(7/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-6*A*b^
3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Se
c[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A +
63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c +
d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*
b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a*(b + a*Cos[c + d*x]
)^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-6*A*b^3
+ 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[
(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 6
3*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*
x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*
(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(35*a^2*Sqrt[b + a*Cos[c + d
*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((-6*A*b^3 + 63*a^3*B + 21*a*b^2*B
+ 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(-6*A*b^3 + 63*a^3*B +
21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^
2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A
+ 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c
+ d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(-6*A*b^3 + 63*a^3*B
+ 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(-6*A*b^3
+ 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)
/2]^2)/2 - ((I/2)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(
c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*C
os[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a
+ b)] + ((I/2)*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSin
h[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((
b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]
^2)/(a + b)] - (a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sec[(c + d*x)/2]^
4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*
Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sec[(c + d
*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b
)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(105*a^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c
+ d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I
*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2
)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSin
h[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a +
b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/
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]))/(35*a^2*Sqrt[b + a*Cos[c + d*x]])))
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fricas [F] time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{3} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{3} + {\left (B a + A b\right )} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*b*cos(d*x + c)^3*sec(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^3*sec(d*x + c)^2 + A*a*cos(d*x + c)^3 +
(B*a + A*b)*cos(d*x + c)^3*sec(d*x + c))*sqrt(b*sec(d*x + c) + a)*sqrt(cos(d*x + c)), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*(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)^(3/2)*cos(d*x + c)^(7/2), x)
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maple [B] time = 2.55, size = 2911, normalized size = 8.11 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-2/105/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^3*(15*A*((a-b)/
(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a^4*(1/(1+cos(d*x+c)))^(3/2)+63*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))
^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^4-63*B*((b+a*cos(d*x+c
))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*
a^4-35*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+c
os(d*x+c))/(a+b))^(1/2)*a^4-6*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)
/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^4-25*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+
c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^4+25*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)
*cos(d*x+c)^2*(1/(1+cos(d*x+c)))^(3/2)*a^4+35*C*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^4*(1/(1+cos(d*x+
c)))^(3/2)+21*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^4*(1/(1+cos(d*x+c)))^(3/2)+21*B*((a-b)/(a+b))^(1
/2)*cos(d*x+c)^2*sin(d*x+c)*a^4*(1/(1+cos(d*x+c)))^(3/2)+63*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(1+cos(d
*x+c)))^(3/2)+42*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+21*B*((a-b)/(a+b))^(1/2)*si
n(d*x+c)*a*b^3*(1/(1+cos(d*x+c)))^(3/2)+35*C*((a-b)/(a+b))^(1/2)*a^3*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3/2)+140
*C*((a-b)/(a+b))^(1/2)*a^2*b^2*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3/2)+63*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)*sin(d*x
+c)*a^4*(1/(1+cos(d*x+c)))^(3/2)+25*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+82*A*((a-b
)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+3*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(1+cos(
d*x+c)))^(3/2)+25*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^4*(1/(1+cos(d*x+c)))^(3/2)+35*C*((a-b)/(a+b))^
(1/2)*sin(d*x+c)*cos(d*x+c)*a^4*(1/(1+cos(d*x+c)))^(3/2)+15*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*(1/(
1+cos(d*x+c)))^(3/2)*a^4+63*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+107*A
*((a-b)/(a+b))^(1/2)*cos(d*x+c)*sin(d*x+c)*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+27*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)*
sin(d*x+c)*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)-3*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)*sin(d*x+c)*a*b^3*(1/(1+cos(d*x+
c)))^(3/2)+63*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)*sin(d*x+c)*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+63*B*((a-b)/(a+b))^(1
/2)*cos(d*x+c)*sin(d*x+c)*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+39*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^
3*b*(1/(1+cos(d*x+c)))^(3/2)+39*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+2
7*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+175*C*((a-b)/(a+b))^(1/2)*cos
(d*x+c)*a^3*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3/2)-105*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+
c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b^2-6*A*((a-b)/(a+b))^(1/2)*sin(d*x
+c)*b^4*(1/(1+cos(d*x+c)))^(3/2)-84*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^
(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^3*b+21*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/
sin(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b^2+63*B*((b+a*cos(d*x+c))/
(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3
*b-21*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c)
,(-(a+b)/(a-b))^(1/2))*a^2*b^2+21*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((
a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+140*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/s
in(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^3*b-140*C*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b
+140*C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),
(-(a+b)/(a-b))^(1/2))*a^2*b^2+82*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/
2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^3*b-51*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin
(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b^2-6*A*EllipticF((-1+cos(d*x+
c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a*b^3-8
2*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(
a+b)/(a-b))^(1/2))*a^3*b+82*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(
a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2+6*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellipti
cE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3)/a^2/((a-b)/(a+b))^(1/2)/(b+a*co
s(d*x+c))/(1/(1+cos(d*x+c)))^(3/2)/sin(d*x+c)^6
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(7/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^(7/2)*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
[Out]
int(cos(c + d*x)^(7/2)*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**(7/2)*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)
[Out]
Timed out
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