\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [822]

Optimal result

Integrand size = 42, antiderivative size = 475 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 b^3 (25 B-49 C)-3 a b^2 (57 B-13 C)-6 a^2 b (3 B-C)+8 a^3 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d} \]

[Out]

2/315*(a-b)*(18*B*a^3*b-246*B*a*b^3-8*C*a^4-33*C*a^2*b^2-147*C*b^4)*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^4/d-2/315*(a-b)*(3*b^3*(25*B-49*C)-3*a*b^2*(57*B-13*C)-6*a^2*b*(3*B-C)+8*a^3*C)*cot(d*x+c)*EllipticF((a+b*se
c(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^3/d-2/315*(18*B*a*b-8*C*a^2-49*C*b^2)*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/b^2/d+2/63*(9*B*b-4*C*
a)*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b^2/d+2/9*C*sec(d*x+c)*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b/d-2/315*(18*B*
a^2*b-75*B*b^3-8*C*a^3-39*C*a*b^2)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4157, 4118, 4167, 4087, 4090, 3917, 4089} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (8 a^3 C-6 a^2 b (3 B-C)-3 a b^2 (57 B-13 C)+3 b^3 (25 B-49 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{315 b^3 d}-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{315 b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{315 b^4 d}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d} \]

[In]

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

[Out]

(2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 33*a^2*b^2*C - 147*b^4*C)*Cot[c + d*x]*EllipticE[
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))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) - 3*a*b^2*(57*B - 13*
C) - 6*a^2*b*(3*B - C) + 8*a^3*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))])/(315*b^3*d) - (2*(18*a
^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*b*B -
8*a^2*C - 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d
*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*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 4087

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Simp[(-B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[Csc[e + f
*x]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 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 4118

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

Rule 4157

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(
x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.), x_Symbol] :> Dist[1/b^2, Int[(a + b*Csc[e + f*x])
^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 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]

Rubi steps \begin{align*} \text {integral}& = \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} (B+C \sec (c+d x)) \, dx \\ & = \frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac {2 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (a C+\frac {7}{2} b C \sec (c+d x)+\frac {1}{2} (9 b B-4 a C) \sec ^2(c+d x)\right ) \, dx}{9 b} \\ & = \frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {3}{4} b (15 b B-2 a C)-\frac {1}{4} \left (18 a b B-8 a^2 C-49 b^2 C\right ) \sec (c+d x)\right ) \, dx}{63 b^2} \\ & = -\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac {8 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{8} b \left (57 a b B-2 a^2 C+49 b^2 C\right )-\frac {3}{8} \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sec (c+d x)\right ) \, dx}{315 b^2} \\ & = -\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac {16 \int \frac {\sec (c+d x) \left (\frac {3}{16} b \left (153 a^2 b B+75 b^3 B+2 a^3 C+186 a b^2 C\right )-\frac {3}{16} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{945 b^2} \\ & = -\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}-\frac {\left ((a-b) \left (3 b^3 (25 B-49 C)-3 a b^2 (57 B-13 C)-6 a^2 b (3 B-C)+8 a^3 C\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b^2}-\frac {\left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b^2} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \cot (c+d x) E\left (\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}}}{315 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 b^3 (25 B-49 C)-3 a b^2 (57 B-13 C)-6 a^2 b (3 B-C)+8 a^3 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\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}}}{315 b^3 d}-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 22.24 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.38 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (2 (a+b) \left (-18 a^3 b B+246 a b^3 B+8 a^4 C+33 a^2 b^2 C+147 b^4 C\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )-2 b (a+b) \left (8 a^3 C-6 a^2 b (3 B+C)+3 a b^2 (57 B+13 C)+3 b^3 (25 B+49 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+\left (-18 a^3 b B+246 a b^3 B+8 a^4 C+33 a^2 b^2 C+147 b^4 C\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 b^3 d (b+a \cos (c+d x))^2 \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x)}+\frac {\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {2 \left (-18 a^3 b B+246 a b^3 B+8 a^4 C+33 a^2 b^2 C+147 b^4 C\right ) \sin (c+d x)}{315 b^3}+\frac {2}{63} \sec ^3(c+d x) (9 b B \sin (c+d x)+10 a C \sin (c+d x))+\frac {2 \sec ^2(c+d x) \left (72 a b B \sin (c+d x)+3 a^2 C \sin (c+d x)+49 b^2 C \sin (c+d x)\right )}{315 b}+\frac {2 \sec (c+d x) \left (9 a^2 b B \sin (c+d x)+75 b^3 B \sin (c+d x)-4 a^3 C \sin (c+d x)+88 a b^2 C \sin (c+d x)\right )}{315 b^2}+\frac {2}{9} b C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))} \]

[In]

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

[Out]

(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8
*a^4*C + 33*a^2*b^2*C + 147*b^4*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)] - 2*b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B
+ C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*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)] + (-18*a^3*b*B + 246*
a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*
x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*(a + b*
Sec[c + d*x])^(3/2)*((2*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^
3) + (2*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(72*a*b*B*Sin[c + d*
x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b) + (2*Sec[c + d*x]*(9*a^2*b*B*Sin[c + d*x] + 75*b^3
*B*Sin[c + d*x] - 4*a^3*C*Sin[c + d*x] + 88*a*b^2*C*Sin[c + d*x]))/(315*b^2) + (2*b*C*Sec[c + d*x]^3*Tan[c + d
*x])/9))/(d*(b + a*Cos[c + d*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5583\) vs. \(2(437)=874\).

Time = 39.27 (sec) , antiderivative size = 5584, normalized size of antiderivative = 11.76

method	result	size
parts	\(\text {Expression too large to display}\)	\(5584\)
default	\(\text {Expression too large to display}\)	\(5647\)

[In]

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

[Out]

result too large to display

Fricas [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{3}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

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

Giac [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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