3.1.0 ∫ (a + b tan(e + fx))3∘__________c + d tan (e + fx)(A + B tan(e + fx) + C tan2(e + fx)) dx [90]

Integrand size = 47, antiderivative size = 464 \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^3 (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]

-(a-I*b)^3*(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/f+(a+I*b)^3*(I*A-B-I*C)*arc

tanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/f+2*(B*a^3-3*B*a*b^2+3*a^2*b*(A-C)-b^3*(A-C))*(c+d*ta

n(f*x+e))^(1/2)/f+2/315*(40*a^3*C*d^3-6*a^2*b*d^2*(-45*B*d+16*C*c)+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(f*x+e))^(3/2)/d^4/f+2/105*b*(21*b*(A*b+B*a-C*b)*d^

2+4*(-a*d+b*c)*(-3*B*b*d-2*C*a*d+2*C*b*c))*tan(f*x+e)*(c+d*tan(f*x+e))^(3/2)/d^3/f-2/21*(-3*B*b*d-2*C*a*d+2*C*

b*c)*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(3/2)/d^2/f+2/9*C*(a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)/d/f

Rubi [A] (veriﬁed)

Time = 2.70 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{315 d^4 f}+\frac {2 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {(a-i b)^3 \sqrt {c-i d} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 \sqrt {c+i d} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{105 d^3 f}-\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]

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]

-(((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))

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,

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

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]

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]

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]

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]

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]

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

Rubi steps \begin{align*} \text {integral}& = \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 ) \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 ((a-i b)^3 (A-i B-C) (i c+d)\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 (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 ) \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 ((i a+b)^3 (A-i B-C) (i c+d)\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) \sqrt {c-i d} \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) \sqrt {c+i d} \text {arctanh}\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] (veriﬁed)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1232\) vs. \(2(464)=928\).

Time = 6.55 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.66 \[ \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 \left (-\frac {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}+\frac {2 \left (\frac {3 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}}{10 d f}-\frac {2 \left (\frac {2 \left (-\frac {15}{8} a d \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 )+b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \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 )\right )\right ) (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {i \left (-\frac {15}{8} a d \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 {3}{4} b c \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 )+\frac {15}{8} a d \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 )+\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \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 {3}{4} 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 )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \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 )\right )\right ) \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 )}{2 f}-\frac {i \left (-\frac {15}{8} a d \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 {3}{4} b c \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 )+\frac {15}{8} a d \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 )-\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \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 {3}{4} 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 )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \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 )\right )\right ) \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 )}{2 f}\right )}{5 d}\right )}{7 d}\right )}{9 d} \]

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]

(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)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(4472\) vs. \(2(424)=848\).

Time = 0.53 (sec) , antiderivative size = 4473, normalized size of antiderivative = 9.64


method	result	size

parts	\(\text {Expression too large to display}\)	\(4473\)
derivativedivides	\(\text {Expression too large to display}\)	\(6661\)
default	\(\text {Expression too large to display}\)	\(6661\)

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,method=_RETURNVERBOSE)

1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^

2)^(1/2)-2*c)^(1/2))*A*a^3+1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(

1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*B*b^3-1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)

^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*A*a^3-1/f*d/(2*(c^2+d^2)^(1/2)-2*c)

^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*B*b^3+1/

f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)

^(1/2)-2*c)^(1/2))*C*a^3-1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/

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

n(f*x+e))^(3/2)-12/5/f/d^3*C*a*b^2*c*(c+d*tan(f*x+e))^(5/2)-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(

2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3+1/4/f/d*ln(d

*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^

(1/2)*a^3*c-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*

(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3+1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2

)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c+1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f

*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^

3-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*C*(2*(c^2+d^

2)^(1/2)+2*c)^(1/2)*a^3*c+2/f/d^3*C*a*b^2*c^2*(c+d*tan(f*x+e))^(3/2)+3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arcta

n((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*C*a*b^2+3/f*d/(2*(c^

2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c

)^(1/2))*A*a*b^2+3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^

(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*B*a^2*b-3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2

*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*C*a*b^2-3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)

*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*A*a*b^2-3/f*d/

(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/

2)-2*c)^(1/2))*B*a^2*b+1/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2

)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3-1/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1

/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+1/4/f/d*ln((c+d*tan(f*x+e

))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^

2)^(1/2)*b^3-1/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*B

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c-1/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x

+e)-c-(c^2+d^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3+1/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(

2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+1/f*a^2*(3*

A*b+B*a)*(2*(c+d*tan(f*x+e))^(1/2)+1/4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1

/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))+((c^2+d^2)^(1/2)-c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c

^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-1/4*(2*(c^2+d^2)^(1/2)+2*c)^

(1/2)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+(-(c^2+d^2)^(1/2

)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2

)^(1/2)-2*c)^(1/2)))+2*C*b^3/f/d^4*(1/9*(c+d*tan(f*x+e))^(9/2)-3/7*c*(c+d*tan(f*x+e))^(7/2)+3/5*c^2*(c+d*tan(f

*x+e))^(5/2)-1/5*(c+d*tan(f*x+e))^(5/2)*d^2-1/3*c^3*(c+d*tan(f*x+e))^(3/2)+1/3*c*d^2*(c+d*tan(f*x+e))^(3/2)+(c

+d*tan(f*x+e))^(1/2)*d^4+d^4*(1/8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2

*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))+1/2*((c^2+d^2)^(1/2)-c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^

2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-1/8*(2*(c^2+d^2)^(1/2)+2*c)^(

1/2)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+1/2*(-(c^2+d^2)^(

1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+

d^2)^(1/2)-2*c)^(1/2))))-3/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d

^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+3/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^

2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b-3/4/f/d*ln(d*tan(f*

x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a

^2*b*c-3/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*C*(2*(c

^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2+3/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(

1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-3/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*

(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b

^2+3/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*A*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*a*b^2*c-3/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c

^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b+3/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2

+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+3/4/f/d*ln((c+d

*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/

2)*(c^2+d^2)^(1/2)*a*b^2-3/4/f/d*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d

^2)^(1/2))*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+3/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^

2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2+2/3/f/d^3*B*b^3*c^2

*(c+d*tan(f*x+e))^(3/2)+6/7/f/d^3*C*a*b^2*(c+d*tan(f*x+e))^(7/2)-4/5/f/d^3*B*b^3*c*(c+d*tan(f*x+e))^(5/2)-2/f/

d*C*a*b^2*(c+d*tan(f*x+e))^(3/2)+2/f*b*(A*b^2+3*B*a*b+3*C*a^2)/d^2*(1/5*(c+d*tan(f*x+e))^(5/2)-1/3*(c+d*tan(f*

x+e))^(3/2)*c-(c+d*tan(f*x+e))^(1/2)*d^2-d^2*(1/8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(

c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))+1/2*((c^2+d^2)^(1/2)-c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2

)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-1/8*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+1

/2*(-(c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)

^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))))+2/7/f/d^3*B*b^3*(c+d*tan(f*x+e))^(7/2)-2/3/f/d*B*b^3*(c+d*tan(f*x+e))

^(3/2)+2/3/f/d*C*a^3*(c+d*tan(f*x+e))^(3/2)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35153 vs. \(2 (414) = 828\).

Time = 10.87 (sec) , antiderivative size = 35153, normalized size of antiderivative = 75.76 \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]

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")

Sympy [F]

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

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)

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)

Maxima [F(-1)]

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

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")

Giac [F(-1)]

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

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")

Mupad [F(-1)]

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

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

\(\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [90]

Optimal result