3.2.3 \(\int \frac {(c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^3} \, dx\) [103]

Optimal. Leaf size=532 \[ -\frac {(A-i B-C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}-\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2} \]

[Out]

-(A-I*B-C)*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)^3/f+(A+I*B-C)*(c+I*d)^(3/2)*arc
tanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f-1/4*(a^5*b*B*d^2+3*a^6*C*d^2+a^4*b^2*d*(4*B*c+3*(A+2*C)
*d)-b^6*(8*A*c^2-3*A*d^2-12*B*c*d-8*C*c^2)+a^2*b^4*(24*A*c^2-26*A*d^2-48*B*c*d-24*C*c^2+35*C*d^2)-2*a^3*b^3*(1
2*c*(A-C)*d+B*(4*c^2-9*d^2))+a*b^5*(40*c*(A-C)*d+3*B*(8*c^2-5*d^2)))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-
a*d+b*c)^(1/2))/b^(5/2)/(a^2+b^2)^3/f/(-a*d+b*c)^(1/2)-1/4*(a^3*b*B*d+3*a^4*C*d+b^4*(3*A*d+4*B*c)+a*b^3*(8*A*c
-7*B*d-8*C*c)-a^2*b^2*(5*A*d+4*B*c-11*C*d))*(c+d*tan(f*x+e))^(1/2)/b^2/(a^2+b^2)^2/f/(a+b*tan(f*x+e))-1/2*(A*b
^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(3/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^2

________________________________________________________________________________________

Rubi [A]
time = 2.85, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3726, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{4 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} f \left (a^2+b^2\right )^3 \sqrt {b c-a d}}-\frac {(c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((A + I*B
- C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) - ((a^5*b*B*d^2 + 3*a^6
*C*d^2 + a^4*b^2*d*(4*B*c + 3*(A + 2*C)*d) - b^6*(8*A*c^2 - 8*c^2*C - 12*B*c*d - 3*A*d^2) + a^2*b^4*(24*A*c^2
- 24*c^2*C - 48*B*c*d - 26*A*d^2 + 35*C*d^2) - 2*a^3*b^3*(12*c*(A - C)*d + B*(4*c^2 - 9*d^2)) + a*b^5*(40*c*(A
- C)*d + 3*B*(8*c^2 - 5*d^2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*b^(5/2)*(a^2 +
b^2)^3*Sqrt[b*c - a*d]*f) - ((a^3*b*B*d + 3*a^4*C*d + b^4*(4*B*c + 3*A*d) + a*b^3*(8*A*c - 8*c*C - 7*B*d) - a
^2*b^2*(4*B*c + 5*A*d - 11*C*d))*Sqrt[c + d*Tan[e + f*x]])/(4*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*
b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)

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 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 3715

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

Rule 3726

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

Rule 3734

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

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} \left (2 (b B-a C) \left (2 b c-\frac {3 a d}{2}\right )+2 A b \left (2 a c+\frac {3 b d}{2}\right )\right )-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac {1}{2} \left (A b^2-a b B-3 a^2 C-4 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {\frac {1}{4} \left (b (2 a c+b d) \left (3 a^2 C d+b^2 (4 B c+3 A d)+a b (4 A c-4 c C-3 B d)\right )-(2 b c-a d) \left (a^2 b B d+3 a^3 C d+A b^2 (4 b c-5 a d)-4 b^3 (c C+B d)-4 a b^2 (B c-2 C d)\right )\right )+2 b^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac {1}{4} d \left (a^3 b B d+3 a^4 C d-a b^3 (8 A c-8 c C-9 B d)-b^4 (4 B c+5 A d-8 C d)+a^2 b^2 (4 B c+3 (A+C) d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {-2 b^2 \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 )+2 b^2 \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}{2 b^2 \left (a^2+b^2\right )^3}+\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{8 b^2 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\left ((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}{2 (a-i b)^3}+\frac {\left ((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}{2 (a+i b)^3}+\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b^2 \left (a^2+b^2\right )^3 f}\\ &=-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\left ((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 (i a+b)^3 f}-\frac {\left ((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 (i a-b)^3 f}+\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 b^2 \left (a^2+b^2\right )^3 d f}\\ &=-\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left ((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 )}{(a-i b)^3 d f}-\frac {\left ((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 )}{(a+i b)^3 d f}\\ &=-\frac {(A-i B-C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}-\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(7678\) vs. \(2(532)=1064\).
time = 6.35, size = 7678, normalized size = 14.43 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4968\) vs. \(2(492)=984\).
time = 0.65, size = 4969, normalized size = 9.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4969\)
default	\(\text {Expression too large to display}\)	\(4969\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*d^2*(1/d^2/(a^2+b^2)^3*(1/4/d*(1/2*(3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2-2*A*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b^3*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b^3*c+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2+2*C*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d+3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a^2*b*c^2-3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-A*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d-3*A*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d-B*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*b^3*c^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2+C*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*
d^2+6*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d+3*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d-3*C*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-6*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*A*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+6*A*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c-3*B*(c^2+d^2)^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*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))+2*(-2*A*(c^2+d^2)^(1/2)*a^3*d^2+6*A*(c^2+d^2)^(1/2)*a^2*b*c*d+6*A*(c^2+d^2)^(1/2)*a*b^2
*d^2-2*A*(c^2+d^2)^(1/2)*b^3*c*d-2*B*(c^2+d^2)^(1/2)*a^3*c*d-6*B*(c^2+d^2)^(1/2)*a^2*b*d^2+6*B*(c^2+d^2)^(1/2)
*a*b^2*c*d+2*B*(c^2+d^2)^(1/2)*b^3*d^2+2*C*(c^2+d^2)^(1/2)*a^3*d^2-6*C*(c^2+d^2)^(1/2)*a^2*b*c*d-6*C*(c^2+d^2)
^(1/2)*a*b^2*d^2+2*C*(c^2+d^2)^(1/2)*b^3*c*d-1/2*(3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2-2*A*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b^3*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b^3*c+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^
2+2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d+3*B*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a^2*b*c^2-3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a^3*c-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
b^3*d-3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d-B*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*
c^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a^3*d^2+6*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d+3*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^
2*b*d-3*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-6*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*
A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^
2*c+6*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c-3*B*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(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)))+1/4/d*(1
/2*(-3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d-B*(c^2+d^2)^(1/2)*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c-3*C*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*b^2*c^2+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d+2
*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d-3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2+3*B*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*a^2*b*d^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a^3*c-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d+3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2
*c^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2-B*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b^3*d^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2-A
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2-6*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*a*b^2*c*d-3*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a*b^2*c+6*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d+3*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*a^2*b*d-3*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-6*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c
*d+3*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+3*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a*b^2*d)*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1...

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 )}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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