\(\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [97]

Optimal result

Integrand size = 47, antiderivative size = 550 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {(i a+b)^3 (A-i B-C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]

[Out]

(I*a+b)^3*(A-I*B-C)*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+(a+I*b)^3*(I*A-B-I*C)*(c+I*d
)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/f+2*(3*a^2*b*(A*c-B*d-C*c)-b^3*(A*c-B*d-C*c)+a^3*(B*c+(A
-C)*d)-3*a*b^2*(B*c+(A-C)*d))*(c+d*tan(f*x+e))^(1/2)/f+2/3*(B*a^3-3*B*a*b^2+3*a^2*b*(A-C)-b^3*(A-C))*(c+d*tan(
f*x+e))^(3/2)/f+2/3465*(168*a^3*C*d^3-2*a^2*b*d^2*(-847*B*d+192*C*c)+33*a*b^2*d*(8*c^2*C-18*B*c*d+63*(A-C)*d^2
)-b^3*(48*c^3*C-88*B*c^2*d+198*c*(A-C)*d^2+693*B*d^3))*(c+d*tan(f*x+e))^(5/2)/d^4/f+2/693*b*(99*b*(A*b+B*a-C*b
)*d^2+4*(-a*d+b*c)*(-11*B*b*d-6*C*a*d+6*C*b*c))*tan(f*x+e)*(c+d*tan(f*x+e))^(5/2)/d^3/f-2/99*(-11*B*b*d-6*C*a*
d+6*C*b*c)*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)/d^2/f+2/11*C*(a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(5/2)/d/
f

Rubi [A] (verified)

Time = 2.85 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {3728, 3718, 3711, 3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{3465 d^4 f}+\frac {2 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac {(a+i b)^3 (c+i d)^{3/2} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {(b+i a)^3 (c-i d)^{3/2} (A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{693 d^3 f}-\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]

[In]

Int[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

((I*a + b)^3*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((a + I*b)^3*(
I*A - B - I*C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(3*a^2*b*(A*c - c*C - B
*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Sqrt[c + d*Tan[e + f*x]])/f +
(2*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(168*a^3*C*d^3
- 2*a^2*b*d^2*(192*c*C - 847*B*d) + 33*a*b^2*d*(8*c^2*C - 18*B*c*d + 63*(A - C)*d^2) - b^3*(48*c^3*C - 88*B*c^
2*d + 198*c*(A - C)*d^2 + 693*B*d^3))*(c + d*Tan[e + f*x])^(5/2))/(3465*d^4*f) + (2*b*(99*b*(A*b + a*B - b*C)*
d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(693*d^3*f) - (2*
(6*b*c*C - 11*b*B*d - 6*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2))/(99*d^2*f) + (2*C*(a + b*Tan
[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2))/(11*d*f)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[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] && GtQ[m, 0]

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3711

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

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac {2 \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} (-6 b c C+a (11 A-5 C) d)+\frac {11}{2} (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (6 b c C-11 b B d-6 a C d) \tan ^2(e+f x)\right ) \, dx}{11 d} \\ & = -\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac {4 \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (\frac {1}{4} \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac {99}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {1}{4} \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{99 d^2} \\ & = \frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {8 \int (c+d \tan (e+f x))^{3/2} \left (\frac {1}{8} \left (-21 a^3 (33 A-25 C) d^3-66 a b^2 c d (4 c C-9 B d)+a^2 b d^2 (384 c C+385 B d)+2 b^3 c \left (24 c^2 C-44 B c d+99 (A-C) d^2\right )\right )-\frac {693}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-\frac {1}{8} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{693 d^3} \\ & = \frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {8 \int (c+d \tan (e+f x))^{3/2} \left (\frac {693}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac {693}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)\right ) \, dx}{693 d^3} \\ & = \frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {8 \int \sqrt {c+d \tan (e+f x)} \left (-\frac {693}{8} d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )-\frac {693}{8} d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{693 d^3} \\ & = \frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {8 \int \frac {\frac {693}{8} d^3 \left (a^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+\frac {693}{8} d^3 \left (3 a^2 b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{693 d^3} \\ & = \frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac {1}{2} \left ((a-i b)^3 (A-i B-C) (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B-C) (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac {\left ((a-i b)^3 (i A+B-i C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left (i (a+i b)^3 (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f} \\ & = \frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\left ((a-i b)^3 (A-i B-C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b)^3 (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f} \\ & = -\frac {(a-i b)^3 (i A+B-i C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (A+i B-C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1290\) vs. \(2(550)=1100\).

Time = 6.57 (sec) , antiderivative size = 1290, normalized size of antiderivative = 2.35 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac {2 \left (\frac {(-6 b c C+11 b B d+6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}+\frac {2 \left (\frac {b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{14 d f}-\frac {2 \left (\frac {2 \left (-\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {i \left (-\frac {7}{8} a d \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac {1}{4} b c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )-\frac {1}{4} b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac {2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}-\frac {i \left (-\frac {7}{8} a d \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac {1}{4} b c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )-\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )-\frac {1}{4} b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac {2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}\right )}{7 d}\right )}{9 d}\right )}{11 d} \]

[In]

Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

(2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2))/(11*d*f) + (2*(((-6*b*c*C + 11*b*B*d + 6*a*C*d)*(a + b
*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2))/(9*d*f) + (2*((b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b
*c*C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(14*d*f) - (2*((2*((-7*a*d*(99*b*(A*b + a
*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^
3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4))*(c + d*Tan[e + f*x]
)^(5/2))/(5*d*f) + ((I/2)*((-7*a*d*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*
d)))/8 + (b*c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4 + (7*a*d*(99*b*(A
*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + ((7*I)/2)*d*((99*a*(a^2*B - b^2*B + 2
*a*b*(A - C))*d^2)/4 + (b*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/4 -
(b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4) - b*((-693*(a^2*B - b^2*B +
2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4))*(
(2*(c + d*Tan[e + f*x])^(3/2))/3 + (c - I*d)*((2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d
]])/(-c + I*d) + 2*Sqrt[c + d*Tan[e + f*x]])))/f - ((I/2)*((-7*a*d*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C -
11*B*d) - a*b*d*(48*c*C + 55*B*d)))/8 + (b*c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d
- 6*a*C*d)))/4 + (7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 - ((7*I
)/2)*d*((99*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (b*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d)
- a*b*d*(48*c*C + 55*B*d)))/4 - (b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d))
)/4) - b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c
*C - 11*b*B*d - 6*a*C*d)))/4))*((2*(c + d*Tan[e + f*x])^(3/2))/3 + (c + I*d)*((2*(c + I*d)^(3/2)*ArcTanh[Sqrt[
c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(-c - I*d) + 2*Sqrt[c + d*Tan[e + f*x]])))/f))/(7*d)))/(9*d)))/(11*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10951\) vs. \(2(507)=1014\).

Time = 0.34 (sec) , antiderivative size = 10952, normalized size of antiderivative = 19.91

method	result	size
parts	\(\text {Expression too large to display}\)	\(10952\)
derivativedivides	\(\text {Expression too large to display}\)	\(11056\)
default	\(\text {Expression too large to display}\)	\(11056\)

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84950 vs. \(2 (498) = 996\).

Time = 194.27 (sec) , antiderivative size = 84950, normalized size of antiderivative = 154.45 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

integrate((a+b*tan(f*x+e))**3*(c+d*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Integral((a + b*tan(e + f*x))**3*(c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)

Maxima [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]

[In]

int((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2),x)

[Out]

\text{Hanged}