3.716 \(\int \cos ^4(c+d x) \sqrt{a+b \sec (c+d x)} (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=587 \[ -\frac{\sqrt{a+b} \left (-4 a^2 b (7 A+12 C)-24 a^3 (3 A+4 C)+10 a A b^2-15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{192 a^3 d}+\frac{b \left (4 a^2 (7 A+12 C)+15 A b^2\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 a^2 d}+\frac{(a-b) \sqrt{a+b} \left (4 a^2 (7 A+12 C)+15 A b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^3 d}+\frac{\sqrt{a+b} \left (8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)+5 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^4 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 a d} \]

[Out]

((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sq
rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(19
2*a^3*d) - (Sqrt[a + b]*(10*a*A*b^2 - 15*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(7*A + 12*C))*Cot[c + d*x]*Ellip
ticF[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))])/(192*a^3*d) + (Sqrt[a + b]*(5*A*b^4 + 8*a^2*b^2*(A + 2*C) - 16*a^4*(3*A +
4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(64*a^4*d) + (b*(15*A*b^2 + 4*a^2*(7*A
+ 12*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d) - ((5*A*b^2 - 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sq
rt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a^2*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(
24*a*d) + (A*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.57532, antiderivative size = 587, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4095, 4104, 4058, 3921, 3784, 3832, 4004} \[ \frac{b \left (4 a^2 (7 A+12 C)+15 A b^2\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 a^2 d}-\frac{\sqrt{a+b} \left (-4 a^2 b (7 A+12 C)-24 a^3 (3 A+4 C)+10 a A b^2-15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^3 d}+\frac{(a-b) \sqrt{a+b} \left (4 a^2 (7 A+12 C)+15 A b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^3 d}+\frac{\sqrt{a+b} \left (8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)+5 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^4 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]

[Out]

((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sq
rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(19
2*a^3*d) - (Sqrt[a + b]*(10*a*A*b^2 - 15*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(7*A + 12*C))*Cot[c + d*x]*Ellip
ticF[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))])/(192*a^3*d) + (Sqrt[a + b]*(5*A*b^4 + 8*a^2*b^2*(A + 2*C) - 16*a^4*(3*A +
4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(64*a^4*d) + (b*(15*A*b^2 + 4*a^2*(7*A
+ 12*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d) - ((5*A*b^2 - 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sq
rt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a^2*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(
24*a*d) + (A*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)

Rule 4095

Int[((A_.) + 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] - Dis
t[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - 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, 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 4058

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, 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, C}, x] && NeQ[a^2 - b^2, 0
]

Rule 3921

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 3784

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(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)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

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 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin{align*} \int \cos ^4(c+d x) \sqrt{a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{4} \int \frac{\cos ^3(c+d x) \left (\frac{A b}{2}+a (3 A+4 C) \sec (c+d x)+\frac{1}{2} b (5 A+8 C) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac{\int \frac{\cos ^2(c+d x) \left (\frac{1}{4} \left (5 A b^2-12 a^2 (3 A+4 C)\right )-\frac{1}{2} a b (17 A+24 C) \sec (c+d x)-\frac{3}{4} A b^2 \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{12 a}\\ &=-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a^2 d}+\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{\int \frac{\cos (c+d x) \left (\frac{1}{8} \left (15 A b^3+8 a^2 \left (\frac{7 A b}{2}+6 b C\right )\right )+\frac{1}{4} a \left (A b^2+12 a^2 (3 A+4 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (5 A b^2-12 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}\\ &=\frac{b \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a^2 d}+\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac{\int \frac{\frac{3}{16} \left (5 A b^4+8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )+\frac{1}{8} a b \left (5 A b^2-12 a^2 (3 A+4 C)\right ) \sec (c+d x)+\frac{1}{16} b^2 \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^3}\\ &=\frac{b \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a^2 d}+\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac{\int \frac{\frac{3}{16} \left (5 A b^4+8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )+\left (\frac{1}{8} a b \left (5 A b^2-12 a^2 (3 A+4 C)\right )-\frac{1}{16} b^2 \left (15 A b^2+4 a^2 (7 A+12 C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^3}-\frac{\left (b^2 \left (15 A b^2+4 a^2 (7 A+12 C)\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^3}\\ &=\frac{(a-b) \sqrt{a+b} \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{192 a^3 d}+\frac{b \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a^2 d}+\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac{\left (5 A b^4+8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{128 a^3}-\frac{\left (b \left (10 a A b^2-15 A b^3-24 a^3 (3 A+4 C)-4 a^2 b (7 A+12 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^3}\\ &=\frac{(a-b) \sqrt{a+b} \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{192 a^3 d}-\frac{\sqrt{a+b} \left (10 a A b^2-15 A b^3-24 a^3 (3 A+4 C)-4 a^2 b (7 A+12 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\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}}}{192 a^3 d}+\frac{\sqrt{a+b} \left (5 A b^4+8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\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}}}{64 a^4 d}+\frac{b \left (15 A b^2+4 a^2 (7 A+12 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^3 d}-\frac{\left (5 A b^2-12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a^2 d}+\frac{A b \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac{A \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}\\ \end{align*}

Mathematica [B]  time = 15.3776, size = 1845, normalized size = 3.14 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]

[Out]

(Sqrt[a + b*Sec[c + d*x]]*((A*b*Sin[c + d*x])/(96*a) + ((48*a^2*A - 5*A*b^2 + 48*a^2*C)*Sin[2*(c + d*x)])/(192
*a^2) + (A*b*Sin[3*(c + d*x)])/(96*a) + (A*Sin[4*(c + d*x)])/32))/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b -
a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(28*a^3*A*b*Tan[(c + d*x)/2] + 28*a^2*A
*b^2*Tan[(c + d*x)/2] + 15*a*A*b^3*Tan[(c + d*x)/2] + 15*A*b^4*Tan[(c + d*x)/2] + 48*a^3*b*C*Tan[(c + d*x)/2]
+ 48*a^2*b^2*C*Tan[(c + d*x)/2] - 56*a^3*A*b*Tan[(c + d*x)/2]^3 - 30*a*A*b^3*Tan[(c + d*x)/2]^3 - 96*a^3*b*C*T
an[(c + d*x)/2]^3 + 28*a^3*A*b*Tan[(c + d*x)/2]^5 - 28*a^2*A*b^2*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Tan[(c + d*x)
/2]^5 - 15*A*b^4*Tan[(c + d*x)/2]^5 + 48*a^3*b*C*Tan[(c + d*x)/2]^5 - 48*a^2*b^2*C*Tan[(c + d*x)/2]^5 - 288*a^
4*A*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Ta
n[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*A*b^2*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a
- b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]
+ 30*A*b^4*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a +
b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 384*a^4*C*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]]
, (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a
+ b)] + 96*a^2*b^2*C*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*S
qrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 288*a^4*A*EllipticPi[-1, -ArcSin[Tan[(c +
d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2
+ b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*A*b^2*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Ta
n[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a +
b)] + 30*A*b^4*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c +
d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 384*a^4*C*EllipticPi[-1, -Ar
cSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(
c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*a^2*b^2*C*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b
)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x
)/2]^2)/(a + b)] + b*(a + b)*(15*A*b^2 + 4*a^2*(7*A + 12*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a +
b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)
/2]^2)/(a + b)] - 2*a*(2*a*A*b^2 + 5*A*b^3 + 24*a^3*(3*A + 4*C) - 12*a^2*b*(3*A + 4*C))*EllipticF[ArcSin[Tan[(
c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c +
d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*a^3*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1 + Tan[
(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 +
Tan[(c + d*x)/2]^2)))

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Maple [B]  time = 0.619, size = 3606, normalized size = 6.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x)

[Out]

1/192/d/a^3*(-1+cos(d*x+c))^2*(-288*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^4*sin(d*x+c)-24*A*a^4*cos(d*x+c)^4+7
2*A*a^4*cos(d*x+c)^2-15*A*cos(d*x+c)^2*b^4+30*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(c
os(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*sin(d*x+c)-28*A*(cos(d*x
+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c)
,((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)-15*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3*sin(d*x+c)-72*A*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a
-b)/(a+b))^(1/2))*a^3*b*sin(d*x+c)+4*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)+10*A*(cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(
a+b))^(1/2))*a*b^3*sin(d*x+c)+192*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*sin(d*x+c)-384*C*a^4*(cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c
),-1,((a-b)/(a+b))^(1/2))-44*A*cos(d*x+c)^3*a^3*b-5*A*cos(d*x+c)^3*a*b^3+28*A*cos(d*x+c)^2*a^3*b-30*A*cos(d*x+
c)^2*a^2*b^2+72*A*cos(d*x+c)*a^3*b+28*A*cos(d*x+c)*a^2*b^2-10*A*cos(d*x+c)*a*b^3-48*A*a^4*cos(d*x+c)^6+15*A*co
s(d*x+c)^2*a*b^3+2*A*cos(d*x+c)^4*a^2*b^2-96*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(co
s(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^3*b-48*C*
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/si
n(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^3*b-48*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a
*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*
x+c)*a^2*b^2+96*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi
((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^2*b^2-15*A*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*b^4*sin(d*x+c)+144*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*
EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*sin(d*x+c)+48*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),
-1,((a-b)/(a+b))^(1/2))*a^2*b^2-28*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos
(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b-28*A*cos(d*x+c)
*a^2*b^2*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*Elliptic
E((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))-15*A*cos(d*x+c)*b^3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))
*a-72*A*cos(d*x+c)*a^3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d
*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b+4*A*cos(d*x+c)*a^2*b^2*(cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(
(a-b)/(a+b))^(1/2))+10*A*cos(d*x+c)*b^3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c
)+1))^(1/2)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a-144*C*cos(d*x+c)^3*a^3*b-48
*C*cos(d*x+c)^2*a^2*b^2-56*A*cos(d*x+c)^5*a^3*b+48*C*cos(d*x+c)^2*a^3*b+96*C*cos(d*x+c)*a^3*b+48*C*cos(d*x+c)*
a^2*b^2-288*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^4+30*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c)
,-1,((a-b)/(a+b))^(1/2))*b^4-15*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*
x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^4+144*A*cos(d*x+c)*sin
(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*
x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4+48*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(co
s(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)-28*A*(cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x
+c),((a-b)/(a+b))^(1/2))*a^3*b*sin(d*x+c)+96*C*cos(d*x+c)^2*a^4+192*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)
*sin(d*x+c)*a^4-384*C*a^4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*El
lipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*x+c)-96*C*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*a^3*b*sin(d*x+c)-48*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)
*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b*sin(d*x+c)-48*C*(cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))
*a^2*b^2*sin(d*x+c)+96*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ell
ipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)-96*C*cos(d*x+c)^4*a^4+15*A*cos(d
*x+c)*b^4)*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)/(b+a*cos(d*x+c))/sin(d*x+c)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt{b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)*cos(d*x + c)^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral((C*cos(d*x + c)^4*sec(d*x + c)^2 + A*cos(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), 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)**4*(A+C*sec(d*x+c)**2)*(a+b*sec(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} + A\right )} \sqrt{b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)*cos(d*x + c)^4, x)