3.90 \(\int (a+b \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=464 \[ \frac{2 (c+d \tan (e+f x))^{3/2} \left (-6 a^2 b d^2 (16 c C-45 B d)+40 a^3 C d^3+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )+b^3 \left (-\left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{315 d^4 f}+\frac{2 \left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{105 d^3 f}-\frac{(a-i b)^3 \sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(a+i b)^3 \sqrt{c+i d} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]

[Out]

-(((a - I*b)^3*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b)^
3*(I*A - B - I*C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a^3*B - 3*a*b^2*B + 3
*a^2*b*(A - C) - b^3*(A - C))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(40*a^3*C*d^3 - 6*a^2*b*d^2*(16*c*C - 45*B*d) +
9*a*b^2*d*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2) - b^3*(16*c^3*C - 24*B*c^2*d + 42*c*(A - C)*d^2 + 105*B*d^3))
*(c + d*Tan[e + f*x])^(3/2))/(315*d^4*f) + (2*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d
- 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(105*d^3*f) - (2*(2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b*Ta
n[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(21*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))
/(9*d*f)

________________________________________________________________________________________

Rubi [A]  time = 2.08898, antiderivative size = 464, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.17, Rules used = {3647, 3637, 3630, 3528, 3539, 3537, 63, 208} \[ \frac{2 (c+d \tan (e+f x))^{3/2} \left (-6 a^2 b d^2 (16 c C-45 B d)+40 a^3 C d^3+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )+b^3 \left (-\left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{315 d^4 f}+\frac{2 \left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{105 d^3 f}-\frac{(a-i b)^3 \sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(a+i b)^3 \sqrt{c+i d} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^3*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b)^
3*(I*A - B - I*C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a^3*B - 3*a*b^2*B + 3
*a^2*b*(A - C) - b^3*(A - C))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(40*a^3*C*d^3 - 6*a^2*b*d^2*(16*c*C - 45*B*d) +
9*a*b^2*d*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2) - b^3*(16*c^3*C - 24*B*c^2*d + 42*c*(A - C)*d^2 + 105*B*d^3))
*(c + d*Tan[e + f*x])^(3/2))/(315*d^4*f) + (2*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d
- 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(105*d^3*f) - (2*(2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b*Ta
n[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(21*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))
/(9*d*f)

Rule 3647

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])))

Rule 3637

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 3630

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 3528

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 3539

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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 63

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 208

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

Rubi steps

\begin{align*} \int (a+b \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)} \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))^{3/2}}{9 d f}+\frac{2 \int (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)} \left (-\frac{3}{2} (2 b c C-a (3 A-C) d)+\frac{9}{2} (A b+a B-b C) d \tan (e+f x)-\frac{3}{2} (2 b c C-3 b B d-2 a C d) \tan ^2(e+f x)\right ) \, dx}{9 d}\\ &=-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac{4 \int (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)} \left (\frac{3}{4} \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac{63}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{3}{4} \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{63 d^2}\\ &=\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac{8 \int \sqrt{c+d \tan (e+f x)} \left (-\frac{3}{8} \left (5 a^3 (21 A-13 C) d^3+18 a b^2 c d (4 c C-7 B d)-3 a^2 b d^2 (32 c C+15 B d)-2 b^3 c \left (8 c^2 C-12 B c d+21 (A-C) d^2\right )\right )-\frac{315}{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{3}{8} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{315 d^3}\\ &=\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac{8 \int \sqrt{c+d \tan (e+f x)} \left (\frac{315}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac{315}{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}{315 d^3}\\ &=\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac{8 \int \frac{-\frac{315}{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{315}{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)}{\sqrt{c+d \tan (e+f x)}} \, dx}{315 d^3}\\ &=\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac{1}{2} \left ((a-i b)^3 (A-i B-C) (c-i d)\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)\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx\\ &=\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac{\left (i (a+i b)^3 (A+i B-C) (c+i d)\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}+\frac{\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \operatorname{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 (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac{\left ((a+i b)^3 (A+i B-C) (c+i d)\right ) \operatorname{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 ((i a+b)^3 (A-i B-C) (i c+d)\right ) \operatorname{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) \sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}-\frac{(i a-b)^3 (A+i B-C) \sqrt{c+i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac{2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac{2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}\\ \end{align*}

Mathematica [B]  time = 6.42582, size = 1232, normalized size = 2.66 \[ \frac{2 C (c+d \tan (e+f x))^{3/2} (a+b \tan (e+f x))^3}{9 d f}+\frac{2 \left (\frac{2 \left (\frac{3 b \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{10 d f}-\frac{2 \left (\frac{2 \left (b \left (\frac{3}{4} c \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )-\frac{315}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )-\frac{15}{8} a d \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )\right ) (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac{i \left (-\frac{15}{8} a d \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right )+\frac{3}{4} b c \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )+\frac{15}{8} a d \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )+\frac{5}{2} i d \left (\frac{63}{4} a \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2+\frac{3}{4} b \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right )-\frac{3}{4} b \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )\right )-b \left (\frac{3}{4} c \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )-\frac{315}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )\right ) \left (\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right ) (c-i d)^{3/2}}{i d-c}+2 \sqrt{c+d \tan (e+f x)}\right )}{2 f}-\frac{i \left (-\frac{15}{8} a d \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right )+\frac{3}{4} b c \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )+\frac{15}{8} a d \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )-\frac{5}{2} i d \left (\frac{63}{4} a \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2+\frac{3}{4} b \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right )-\frac{3}{4} b \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )\right )-b \left (\frac{3}{4} c \left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right )-\frac{315}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )\right ) \left (\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right ) (c+i d)^{3/2}}{-c-i d}+2 \sqrt{c+d \tan (e+f x)}\right )}{2 f}\right )}{5 d}\right )}{7 d}-\frac{3 (2 b c C-2 a d C-3 b B d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]*(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])^(3/2))/(9*d*f) + (2*((-3*(2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b
*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(7*d*f) + (2*((3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2
*b*c*C - 3*b*B*d - 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(10*d*f) - (2*((2*((-15*a*d*(21*b*(A*b +
a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 + b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d
^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*(c + d*Tan[e + f*
x])^(3/2))/(3*d*f) + ((I/2)*((-15*a*d*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d
)))/8 + (3*b*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4 + (15*a*d*(21*b*(
A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 + ((5*I)/2)*d*((63*a*(a^2*B - b^2*B + 2
*a*b*(A - C))*d^2)/4 + (3*b*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/4 - (3
*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4) - b*((-315*(a^2*B - b^2*B +
2*a*b*(A - C))*d^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*(
(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)*((-15*a*d*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/8 + (3*b*c*(2
1*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4 + (15*a*d*(21*b*(A*b + a*B - b*C)*
d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 - ((5*I)/2)*d*((63*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2
)/4 + (3*b*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/4 - (3*b*(21*b*(A*b + a
*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4) - b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^
3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*((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))/(5*d)))/(7*d)
))/(9*d)

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Maple [B]  time = 0.227, size = 6661, normalized size = 14.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (e + f x \right )}\right )^{3} \sqrt{c + d \tan{\left (e + f x \right )}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)*(a+b*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{3} \sqrt{d \tan \left (f x + e\right ) + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

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