3.8.17 \(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x)) \, dx\) [717]

Optimal. Leaf size=550 \[ \frac {4 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1155 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)+3 a b^3 (209 A+157 C)\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1155 b^4 d}+\frac {2 \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^3 d}+\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \]

[Out]

4/1155*a*(a-b)*(8*a^4*C+3*a^2*b^2*(11*A+6*C)-b^4*(451*A+348*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a
+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^5/
d+2/1155*(a-b)*(16*a^4*C+12*a^3*b*C+6*a^2*b^2*(11*A+8*C)-25*b^4*(11*A+9*C)+3*a*b^3*(209*A+157*C))*cot(d*x+c)*E
llipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-
b*(1+sec(d*x+c))/(a-b))^(1/2)/b^4/d+2/11*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/1155*(8*a^4*C+25
*b^4*(11*A+9*C)+a^2*b^2*(33*A+19*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d+4/1155*a*(132*A*b^2-3*C*a^2+101*C
*b^2)*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/231*(a^2*C+3*b^2*(11*A+9*C))*sec(d*x+c)^2*(a+b*sec(
d*x+c))^(1/2)*tan(d*x+c)/b/d+2/33*a*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 1.29, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4182, 4181, 4187, 4177, 4167, 4090, 3917, 4089} \begin {gather*} \frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{231 b d}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{1155 b^2 d}+\frac {4 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\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 (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{1155 b^5 d}+\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{1155 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\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 (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{1155 b^4 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{33 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(4*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSi
n[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))])/(1155*b^5*d) + (2*(a - b)*Sqrt[a + b]*(16*a^4*C + 12*a^3*b*C + 6*a^2*b^2*(11*A + 8*C
) - 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Cot[c + d*x]*EllipticF[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))])/(1155
*b^4*d) + (2*(8*a^4*C + 25*b^4*(11*A + 9*C) + a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1
155*b^3*d) + (4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1155*
b^2*d) + (2*(a^2*C + 3*b^2*(11*A + 9*C))*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(231*b*d) + (2*
a*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(33*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3
/2)*Tan[c + d*x])/(11*d)

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4090

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[Csc[e + f*x]*((1 +
Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]

Rule 4167

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m
+ 2))), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*
B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 4177

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^
(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m +
2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C,
m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4181

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 4182

Int[((A_.) + 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
+ b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C
, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&  !LeQ[n, -1]

Rule 4187

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)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(
d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a
*C*n)*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] && GtQ[n, 0]

Rubi steps

\begin {align*} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {2}{11} \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {1}{2} a (11 A+6 C)+\frac {1}{2} b (11 A+9 C) \sec (c+d x)+\frac {3}{2} a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {4}{99} \int \frac {\sec ^3(c+d x) \left (\frac {9}{4} a^2 (11 A+8 C)+\frac {3}{2} a b (33 A+26 C) \sec (c+d x)+\frac {3}{4} \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {8 \int \frac {\sec ^2(c+d x) \left (\frac {3}{2} a \left (a^2 C+3 b^2 (11 A+9 C)\right )+\frac {3}{8} b \left (15 b^2 (11 A+9 C)+a^2 (231 A+173 C)\right ) \sec (c+d x)+\frac {3}{4} a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{693 b}\\ &=\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {16 \int \frac {\sec (c+d x) \left (\frac {3}{4} a^2 \left (132 A b^2-3 a^2 C+101 b^2 C\right )+\frac {3}{8} a b \left (726 A b^2+\left (a^2+573 b^2\right ) C\right ) \sec (c+d x)+\frac {9}{16} \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 b^2}\\ &=\frac {2 \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^3 d}+\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {32 \int \frac {\sec (c+d x) \left (-\frac {9}{32} b \left (4 a^4 C-25 b^4 (11 A+9 C)-3 a^2 b^2 (187 A+141 C)\right )-\frac {9}{16} a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{10395 b^3}\\ &=\frac {2 \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^3 d}+\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {\left ((a-b) \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)+3 a b^3 (209 A+157 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{1155 b^3}-\frac {\left (2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{1155 b^3}\\ &=\frac {4 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 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}}}{1155 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)+3 a b^3 (209 A+157 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}}}{1155 b^4 d}+\frac {2 \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^3 d}+\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3988\) vs. \(2(550)=1100\).
time = 26.47, size = 3988, normalized size = 7.25 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((-8*a*(33*a^2*A*b^2 - 451*A*b^4 + 8*a^4*C +
18*a^2*b^2*C - 348*b^4*C)*Sin[c + d*x])/(1155*b^4) + (4*Sec[c + d*x]^3*(33*A*b^2*Sin[c + d*x] + a^2*C*Sin[c +
d*x] + 27*b^2*C*Sin[c + d*x]))/(231*b) + (8*Sec[c + d*x]^2*(132*a*A*b^2*Sin[c + d*x] - 3*a^3*C*Sin[c + d*x] +
101*a*b^2*C*Sin[c + d*x]))/(1155*b^2) + (4*Sec[c + d*x]*(33*a^2*A*b^2*Sin[c + d*x] + 275*A*b^4*Sin[c + d*x] +
8*a^4*C*Sin[c + d*x] + 19*a^2*b^2*C*Sin[c + d*x] + 225*b^4*C*Sin[c + d*x]))/(1155*b^3) + (16*a*C*Sec[c + d*x]
^3*Tan[c + d*x])/33 + (4*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c +
2*d*x])) + (8*((4*a^3*A)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*A*b)/(105*Sqrt[b + a*Cos[
c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^5*C)/(1155*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (24*a^3*C)
/(385*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (464*a*b*C)/(385*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c +
d*x]]) - (62*a^2*A*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*A*Sqrt[Sec[c + d*x]])/(35*b^2*S
qrt[b + a*Cos[c + d*x]]) + (10*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (26*a^2*C*Sqrt[Sec[c
+ d*x]])/(55*Sqrt[b + a*Cos[c + d*x]]) + (32*a^6*C*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (
64*a^4*C*Sqrt[Sec[c + d*x]])/(1155*b^2*Sqrt[b + a*Cos[c + d*x]]) + (30*b^2*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b +
a*Cos[c + d*x]]) - (164*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*A*C
os[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (464*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c
+ d*x]])/(385*Sqrt[b + a*Cos[c + d*x]]) + (32*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a
*Cos[c + d*x]]) + (24*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(385*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[
(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(8*a^4*C + 3*a^2*b
^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b
)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*
C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x
])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b
)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)
/2]^2]*Sec[c + d*x]^(7/2)*((4*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(8*a^4*C + 3*a
^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a
+ b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^
3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c +
d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a
+ b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c +
d*x)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c
+ d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C
))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[Arc
Sin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*
(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a +
b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A +
6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*
Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (8*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(8*a^4*C + 3
*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*(a +
b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d
*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2
- Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^4*C + 12*a^3*b*C
- 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*
(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Co
s[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (a*(a + b)*(8*a^
4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4695\) vs. \(2(508)=1016\).
time = 1.23, size = 4696, normalized size = 8.54

method	result	size
default	\(\text {Expression too large to display}\)	\(4696\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

2/1155/d*(1+cos(d*x+c))^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(105*C*b^6-696*C*cos(d*x+c)^7*
a^2*b^4-225*C*cos(d*x+c)^7*a*b^5-66*A*cos(d*x+c)^6*a^4*b^2+66*A*cos(d*x+c)^6*a^3*b^3+605*A*cos(d*x+c)^6*a^2*b^
4-902*A*cos(d*x+c)^6*a*b^5+429*A*cos(d*x+c)^3*a*b^5-C*cos(d*x+c)^3*a^3*b^3+92*C*cos(d*x+c)^3*a*b^5+145*C*cos(d
*x+c)^2*a^2*b^4+245*C*cos(d*x+c)*a*b^5+16*C*cos(d*x+c)^6*a^5*b-38*C*cos(d*x+c)^6*a^4*b^2+36*C*cos(d*x+c)^6*a^3
*b^3+475*C*cos(d*x+c)^6*a^2*b^4-696*C*cos(d*x+c)^6*a*b^5-33*A*cos(d*x+c)^5*a^3*b^3+748*A*cos(d*x+c)^5*a*b^5-8*
C*cos(d*x+c)^5*a^5*b-16*C*cos(d*x+c)^5*a^3*b^3+584*C*cos(d*x+c)^5*a*b^5+297*A*cos(d*x+c)^4*a^2*b^4+2*C*cos(d*x
+c)^4*a^4*b^2+76*C*cos(d*x+c)^4*a^2*b^4+66*A*cos(d*x+c)^7*a^4*b^2-33*A*cos(d*x+c)^7*a^3*b^3-902*A*cos(d*x+c)^7
*a^2*b^4-275*A*cos(d*x+c)^7*a*b^5-8*C*cos(d*x+c)^7*a^5*b+36*C*cos(d*x+c)^7*a^4*b^2-19*C*cos(d*x+c)^7*a^3*b^3-2
75*A*cos(d*x+c)^6*b^6+66*A*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+
b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3*b^3-275*A*cos(d*x+c)^6*(cos
(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+
c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^6-225*C*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/
(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^6-16*C*cos(
d*x+c)^6*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos
(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^6-275*A*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a
*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*
b^6-225*C*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellipti
cF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^6-16*C*cos(d*x+c)^5*EllipticE((-1+cos(d*x+c))/
sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2
)*sin(d*x+c)*a^6+902*A*cos(d*x+c)^6*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^2*b^4+902*A*cos(d*x+c)^6*Elliptic
E((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d
*x+c))/(a+b))^(1/2)*sin(d*x+c)*a*b^5+16*C*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+
cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^5*b+4*C*cos(d*
x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c
))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^4*b^2+36*C*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+
a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)
*a^3*b^3-423*C*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*El
lipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^2*b^4-696*C*cos(d*x+c)^6*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a
+b))^(1/2))*sin(d*x+c)*a*b^5-16*C*cos(d*x+c)^6*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^5*b-36*C*cos(d*x+c)^6*
EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/
(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^4*b^2-36*C*cos(d*x+c)^6*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(
a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^3*b^
3+696*C*cos(d*x+c)^6*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/
2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^2*b^4+696*C*cos(d*x+c)^6*EllipticE((-1+cos(d*x+c
))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(
1/2)*sin(d*x+c)*a*b^5+66*A*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+
b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3*b^3-561*A*cos(d*x+c)^5*(cos
(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+
c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^2*b^4-902*A*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+
c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a*b^5-66*
A*cos(d*x+c)^5*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b
+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*sin(d*x+c)*a^4*b^2-66*A*cos(d*x+c)^5*EllipticE((-1+cos(d*x+c))/sin(
d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(...

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b*sec(d*x + c)^6 + C*a*sec(d*x + c)^5 + A*b*sec(d*x + c)^4 + A*a*sec(d*x + c)^3)*sqrt(b*sec(d*x +
c) + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(3/2)*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((A + C*sec(c + d*x)**2)*(a + b*sec(c + d*x))**(3/2)*sec(c + d*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c) + a)^(3/2)*sec(d*x + c)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3,x)

[Out]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3, x)

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