3.1353 \(\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=453 \[ \frac{\left (a^2 b (96 A+59 C)+48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left (30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+6 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\cos (c+d x)}} \]
[Out]
((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Elliptic
F[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5
*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*
d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[
c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a +
b)]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c +
d*x]]) + ((6*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[
c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])
________________________________________________________________________________________
Rubi [A] time = 1.8757, antiderivative size = 453, normalized size of antiderivative = 1.,
number of steps used = 15, number of rules used = 13, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.289, Rules
used = {4265, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}
\[ \frac{\sin (c+d x) \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\left (a^2 b (96 A+59 C)+48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left (30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+6 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Elliptic
F[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5
*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*
d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[
c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a +
b)]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c +
d*x]]) + ((6*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[
c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])
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]
Rule 4096
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[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d
*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^
n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C
*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&
!LeQ[n, -1]
Rule 4108
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a +
b*Csc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; Fre
eQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Rule 3859
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(d*Sqr
t[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f
*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 2807
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Rule 2805
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 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 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 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 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 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 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 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]
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \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))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{3} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} a (6 A-C)+(3 A b+3 a B+2 b C) \sec (c+d x)+\frac{1}{2} (6 b B+5 a C) \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{6} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \sec (c+d x)} \left (\frac{3}{4} a (8 a A-2 b B-3 a C)+\frac{1}{2} \left (12 a^2 B+6 b^2 B+a b (24 A+11 C)\right ) \sec (c+d x)+\frac{1}{4} \left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{6} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac{1}{4} a \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \sec (c+d x)+\frac{3}{8} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{6} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac{1}{4} a \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx+\frac{1}{16} \left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{48} \left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 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+\frac{1}{48} \left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 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+\frac{\left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt{b+a \cos (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{b+a \cos (c+d x)}} \, dx}{16 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}\\ &=\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{\left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt{b+a \cos (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{48 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}}\right ) \int \frac{\sec (c+d x)}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{16 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{48 \sqrt{b+a \cos (c+d x)}}\\ &=\frac{\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{\left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 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}{48 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 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}{48 \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=\frac{\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 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 )}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 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)}}{24 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}+\frac{\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 36.5855, size = 157926, normalized size = 348.62 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
Result too large to show
________________________________________________________________________________________
Maple [C] time = 1.01, size = 3162, normalized size = 7. \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x)
[Out]
-1/24/d*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^3*(66*B*sin(d*x+c)*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a*b^2*(1/(cos(d*x
+c)+1))^(3/2)+59*C*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^2*b*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+34*C*cos(d*x+c)^
2*((a-b)/(a+b))^(1/2)*a*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+34*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^2*sin(
d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+24*A*sin(d*x+c)*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a*b^2*(1/(cos(d*x+c)+1))^(3/2
)+54*B*sin(d*x+c)*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b*(1/(cos(d*x+c)+1))^(3/2)+12*B*sin(d*x+c)*cos(d*x+c)^3
*((a-b)/(a+b))^(1/2)*a*b^2*(1/(cos(d*x+c)+1))^(3/2)+26*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b*sin(d*x+c)*(1/
(cos(d*x+c)+1))^(3/2)+16*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+24*A*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/
(a-b))^(1/2))*cos(d*x+c)^3*a*b^2-144*A*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(
(-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b+48*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)
^3*sin(d*x+c)*a^2*b*(1/(cos(d*x+c)+1))^(3/2)+48*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-
1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a^2*b-48*A*(1/(a+b)*(b+a*cos(d
*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*co
s(d*x+c)^3*a^3-48*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1
/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a^3+96*A*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^2+54*B*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^3*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-54*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^3*EllipticE((-1+cos(
d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^2-120*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*
x+c)^3*a*b^2-26*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2
)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a^2*b+44*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ell
ipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a*b^2-33*C*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^
(1/2))*cos(d*x+c)^3*a^2*b+16*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b
)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a*b^2-240*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*
x+c)^3*a*b^2+48*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2
)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a^3+24*A*sin(d*x+c)*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*b^3*(1/(c
os(d*x+c)+1))^(3/2)+16*C*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+12*B*cos(d*x
+c)^2*sin(d*x+c)*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(3/2)*b^3+12*B*cos(d*x+c)*sin(d*x+c)*((a-b)/(a+b))^(1/
2)*(1/(cos(d*x+c)+1))^(3/2)*b^3+8*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)-180
*B*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^3*(
1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b+36*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*
x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b-12*B*EllipticF((
-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(co
s(d*x+c)+1))^(1/2)*a*b^2-24*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)
/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*b^3+8*C*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)*(1/(cos
(d*x+c)+1))^(3/2)-48*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b)
)^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^3*b^3+24*B*EllipticF((-1+cos(d*x+c))*((a-b)/(
a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3-
30*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c)
,(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^3*a^3-18*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elli
pticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*a^3+33*C*(1/(a+b)*(b+a
*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/
2))*cos(d*x+c)^3*a^3-16*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+
b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^3*b^3+33*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*sin(d*x+
c)*(1/(cos(d*x+c)+1))^(3/2)+48*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*a^3*(1/(cos(d*x+c)+1))^(3/2))*((b
+a*cos(d*x+c))/cos(d*x+c))^(1/2)/((a-b)/(a+b))^(1/2)/(b+a*cos(d*x+c))/sin(d*x+c)^6/cos(d*x+c)^(5/2)/(1/(cos(d*
x+c)+1))^(3/2)
________________________________________________________________________________________
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))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [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))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
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))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)*cos(d*x+c)**(1/2),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 )}^{\frac{5}{2}} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*sqrt(cos(d*x + c)), x)