3.140 \(\int \frac{(c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=586 \[ -\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a^4 b^2 d (d (8 A+C)+10 B c)+2 a^5 b B d^2+3 a^6 C d^2+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{15 b^2 f \left (a^2+b^2\right )^3 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (7 A d+5 B c-13 C d)+2 a^3 b B d+3 a^4 C d+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{15 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{(c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]

[Out]

-(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(2*a^3*b*B*d + 3*a^4*C*d + b^4
*(5*B*c + 3*A*d) + 2*a*b^3*(5*A*c - 5*c*C - 4*B*d) - a^2*b^2*(5*B*c + 7*A*d - 13*C*d))*Sqrt[c + d*Tan[e + f*x]
])/(15*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(2*a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(10*B*c +
(8*A + C)*d) + a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 49*A*d^2 + 58*C*d^2) - a^3*b^3*(50*c*(A - C)*d + B*(
15*c^2 - 39*d^2)) + a*b^5*(70*c*(A - C)*d + B*(45*c^2 - 23*d^2)) + b^6*(5*c*(3*c*C + 4*B*d) - 3*A*(5*c^2 - d^2
)))*Sqrt[c + d*Tan[e + f*x]])/(15*b^2*(a^2 + b^2)^3*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b
*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 3.66806, antiderivative size = 586, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3645, 3649, 3616, 3615, 93, 208} \[ -\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a^4 b^2 d (d (8 A+C)+10 B c)+2 a^5 b B d^2+3 a^6 C d^2+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{15 b^2 f \left (a^2+b^2\right )^3 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (7 A d+5 B c-13 C d)+2 a^3 b B d+3 a^4 C d+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{15 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{(c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]

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])^(7/2),x]

[Out]

-(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(2*a^3*b*B*d + 3*a^4*C*d + b^4
*(5*B*c + 3*A*d) + 2*a*b^3*(5*A*c - 5*c*C - 4*B*d) - a^2*b^2*(5*B*c + 7*A*d - 13*C*d))*Sqrt[c + d*Tan[e + f*x]
])/(15*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(2*a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(10*B*c +
(8*A + C)*d) + a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 49*A*d^2 + 58*C*d^2) - a^3*b^3*(50*c*(A - C)*d + B*(
15*c^2 - 39*d^2)) + a*b^5*(70*c*(A - C)*d + B*(45*c^2 - 23*d^2)) + b^6*(5*c*(3*c*C + 4*B*d) - 3*A*(5*c^2 - d^2
)))*Sqrt[c + d*Tan[e + f*x]])/(15*b^2*(a^2 + b^2)^3*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b
*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))

Rule 3645

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*T
an[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 3649

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*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3616

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

Rule 3615

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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 \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))^{7/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{2 \int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} ((b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d))-\frac{5}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac{1}{2} \left (2 A b^2-2 a b B-3 a^2 C-5 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx}{5 b \left (a^2+b^2\right )}\\ &=-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{4 \int \frac{\frac{1}{4} \left (b (3 a c+b d) ((b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d))-(3 b c-a d) \left (2 a^2 b B d+3 a^3 C d+A b^2 (5 b c-7 a d)-5 b^3 (c C+B d)-5 a b^2 (B c-2 C d)\right )\right )+\frac{15}{4} 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 (2 a^3 b B d+3 a^4 C d-2 a b^3 (10 A c-10 c C-11 B d)-b^4 (10 B c+3 (4 A-5 C) d)+2 a^2 b^2 (5 B c+4 A d-C d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{8 \int \frac{\frac{15}{8} b^2 (b c-a d) \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{15}{8} b^2 (b c-a d) \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{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d)}\\ &=-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\left ((A-i B-C) (c-i d)^2\right ) \int \frac{1+i \tan (e+f x)}{\sqrt{a+b \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{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}\\ &=-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\left ((A-i B-C) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^3 f}+\frac{\left ((A+i B-C) (c+i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^3 f}\\ &=-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\left ((A-i B-C) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{i a+b-(i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^3 f}+\frac{\left ((A+i B-C) (c+i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^3 f}\\ &=-\frac{(i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}-\frac{(B-i (A-C)) (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac{2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\\ \end{align*}

Mathematica [B]  time = 9.00577, size = 3134, normalized size = 5.35 \[ \text{Result too large to show} \]

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])^(7/2),x]

[Out]

-((C*(c + d*Tan[e + f*x])^(3/2))/(b*f*(a + b*Tan[e + f*x])^(5/2))) - (-((3*b*c*C - 2*b*B*d - 3*a*C*d)*Sqrt[c +
d*Tan[e + f*x]])/(4*b*f*(a + b*Tan[e + f*x])^(5/2)) - ((-2*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C
- B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*
C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*(b*c - a*d)*f*(a +
b*Tan[e + f*x])^(5/2)) - (2*((-2*(b^2*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*
(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*
d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) - a*((5*b*(b*c - a*d)*((b*(8*A*
b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b
*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b
^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-(a*(8*b^2*d*(B*c + (A - C)*d) +
(b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))))*Sqrt[c + d*Tan[e +
f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-15*b^2*(b*c - a*d)^2*(((-3*a^2*A*b*c^2
+ A*b^3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C + 2*a^3*A*c*d - 6*a*A*b^2*c*d + 6*a^2*b*B
*c*d - 2*b^3*B*c*d - 2*a^3*c*C*d + 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2 - a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a
^2*b*C*d^2 + b^3*C*d^2 + I*(-(a^3*A*c^2) + 3*a*A*b^2*c^2 - 3*a^2*b*B*c^2 + b^3*B*c^2 + a^3*c^2*C - 3*a*b^2*c^2
*C - 6*a^2*A*b*c*d + 2*A*b^3*c*d + 2*a^3*B*c*d - 6*a*b^2*B*c*d + 6*a^2*b*c*C*d - 2*b^3*c*C*d + a^3*A*d^2 - 3*a
*A*b^2*d^2 + 3*a^2*b*B*d^2 - b^3*B*d^2 - a^3*C*d^2 + 3*a*b^2*C*d^2))*ArcTan[(Sqrt[-c - I*d]*Sqrt[a + b*Tan[e +
f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[-c - I*d]) + ((3*a^2*A*b*c^2 - A*b^3*c^
2 - a^3*B*c^2 + 3*a*b^2*B*c^2 - 3*a^2*b*c^2*C + b^3*c^2*C - 2*a^3*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^
3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d - 3*a^2*A*b*d^2 + A*b^3*d^2 + a^3*B*d^2 - 3*a*b^2*B*d^2 + 3*a^2*b*C*d^2
- b^3*C*d^2 + I*(-(a^3*A*c^2) + 3*a*A*b^2*c^2 - 3*a^2*b*B*c^2 + b^3*B*c^2 + a^3*c^2*C - 3*a*b^2*c^2*C - 6*a^2*
A*b*c*d + 2*A*b^3*c*d + 2*a^3*B*c*d - 6*a*b^2*B*c*d + 6*a^2*b*c*C*d - 2*b^3*c*C*d + a^3*A*d^2 - 3*a*A*b^2*d^2
+ 3*a^2*b*B*d^2 - b^3*B*d^2 - a^3*C*d^2 + 3*a*b^2*C*d^2))*ArcTan[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqr
t[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[c - I*d])))/(2*(a^2 + b^2)*f) - (2*(b^2*((b^2*d -
(3*a*(b*c - a*d))/2)*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5
*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b
*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) + ((-3*b*c)/2 + (a*d)/2)*((5*b*(b*c - a*d)*((b*(
8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d)
+ (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*a*d*((b^2*(8
*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-(a*(8*b^2*d*(B*c + (A - C)*
d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))))) - a*((3*b*(b*c
- a*d)*((-5*a*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 -
(b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c
^2 - d^2))))/2 - 2*b*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 -
a*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(
c^2 - d^2)))) + b*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2
*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d
- 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))))/2 - a*d*(b^2*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*
A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-(a*(8*b
^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))
)) - a*((5*b*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 -
(b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^
2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a
*(-(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 + 2*b^3*(2*c*(A - C)*d + B*(c
^2 - d^2))))))))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^
2)*(b*c - a*d))))/(5*(a^2 + b^2)*(b*c - a*d)))/(2*b))/b

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{(A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2}) \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

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))^(7/2),x)

[Out]

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))^(7/2),x)

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

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))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**(7/2),x)

[Out]

Timed out

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

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))^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError