3.104 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=503 \[ \frac{2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{693 d^3 f}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (a^2 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}+\frac{2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{3 f}-\frac{(a-i b)^2 (c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(a+i b)^2 (c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f} \]

[Out]

-(((a - I*b)^2*(I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b
)^2*(I*A - B - I*C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f - (2*(2*a*b*(c^2*C + 2*
B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sq
rt[c + d*Tan[e + f*x]])/f + (2*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c +
d*Tan[e + f*x])^(3/2))/(3*f) + (2*(a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^(5/2))/(5*f) + (2*(36*a
^2*C*d^2 - 22*a*b*d*(2*c*C - 9*B*d) + b^2*(8*c^2*C - 22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(
693*d^3*f) - (2*b*(4*b*c*C - 11*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(99*d^2*f) + (2*C*(a
+ b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f)

________________________________________________________________________________________

Rubi [A]  time = 2.31229, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ \frac{2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{693 d^3 f}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (a^2 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}+\frac{2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{3 f}-\frac{(a-i b)^2 (c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(a+i b)^2 (c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

-(((a - I*b)^2*(I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b
)^2*(I*A - B - I*C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f - (2*(2*a*b*(c^2*C + 2*
B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sq
rt[c + d*Tan[e + f*x]])/f + (2*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c +
d*Tan[e + f*x])^(3/2))/(3*f) + (2*(a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^(5/2))/(5*f) + (2*(36*a
^2*C*d^2 - 22*a*b*d*(2*c*C - 9*B*d) + b^2*(8*c^2*C - 22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(
693*d^3*f) - (2*b*(4*b*c*C - 11*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(99*d^2*f) + (2*C*(a
+ b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*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))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac{2 \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left (\frac{1}{2} (-4 b c C+a (11 A-7 C) d)+\frac{11}{2} (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (4 b c C-11 b B d-4 a C d) \tan ^2(e+f x)\right ) \, dx}{11 d}\\ &=-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{4 \int (c+d \tan (e+f x))^{5/2} \left (\frac{1}{4} \left (44 a b c C d-9 a^2 (11 A-7 C) d^2-4 b^2 \left (2 c^2 C-\frac{11 B c d}{2}\right )\right )-\frac{99}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)-\frac{1}{4} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) \tan ^2(e+f x)\right ) \, dx}{99 d^2}\\ &=\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{4 \int (c+d \tan (e+f x))^{5/2} \left (\frac{99}{4} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2-\frac{99}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{4 \int (c+d \tan (e+f x))^{3/2} \left (-\frac{99}{4} d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-\frac{99}{4} d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{4 \int \sqrt{c+d \tan (e+f x)} \left (\frac{99}{4} d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+\frac{99}{4} d^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)\right ) \, dx}{99 d^2}\\ &=-\frac{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 ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{4 \int \frac{-\frac{99}{4} d^2 \left (a^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+\frac{99}{4} d^2 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{99 d^2}\\ &=-\frac{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 ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac{1}{2} \left ((a-i b)^2 (A-i B-C) (c-i d)^3\right ) \int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx+\frac{1}{2} \left ((a+i b)^2 (A+i B-C) (c+i d)^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx\\ &=-\frac{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 ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac{\left ((a-i b)^2 (i A+B-i C) (c-i d)^3\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 (i (a+i b)^2 (A+i B-C) (c+i d)^3\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 (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 ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac{\left ((a-i b)^2 (A-i B-C) (c-i d)^3\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 ((a+i b)^2 (A+i B-C) (c+i d)^3\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)^2 (i A+B-i C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}-\frac{(a+i b)^2 (B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{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 ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac{2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}\\ \end{align*}

Mathematica [A]  time = 6.43647, size = 564, normalized size = 1.12 \[ \frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac{2 \left (\frac{b \tan (e+f x) (4 a C d+11 b B d-4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \left (\frac{(c+d \tan (e+f x))^{7/2} \left (-36 a^2 C d^2+22 a b d (2 c C-9 B d)+b^2 \left (-\left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )\right )}{14 d f}+\frac{i \left (\frac{99}{4} d^2 \left (a^2 (-(A-C))+2 a b B+b^2 (A-C)\right )+\frac{99}{4} i d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right )\right ) \left (\frac{2}{5} (c+d \tan (e+f x))^{5/2}+(c-i d) \left (\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (2 \sqrt{c+d \tan (e+f x)}+\frac{2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{-c+i d}\right )\right )\right )}{2 f}-\frac{i \left (\frac{99}{4} d^2 \left (a^2 (-(A-C))+2 a b B+b^2 (A-C)\right )-\frac{99}{4} i d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right )\right ) \left (\frac{2}{5} (c+d \tan (e+f x))^{5/2}+(c+i d) \left (\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (2 \sqrt{c+d \tan (e+f x)}+\frac{2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{-c-i d}\right )\right )\right )}{2 f}\right )}{9 d}\right )}{11 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

(2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f) + (2*((b*(-4*b*c*C + 11*b*B*d + 4*a*C*d)*Tan[
e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(9*d*f) - (2*(((-36*a^2*C*d^2 + 22*a*b*d*(2*c*C - 9*B*d) - b^2*(8*c^2*C -
22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(14*d*f) + ((I/2)*(((99*I)/4)*(a^2*B - b^2*B + 2*a*b*
(A - C))*d^2 + (99*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (c - I*d)
*((2*(c + d*Tan[e + f*x])^(3/2))/3 + (c - I*d)*((2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I
*d]])/(-c + I*d) + 2*Sqrt[c + d*Tan[e + f*x]]))))/f - ((I/2)*(((-99*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2
+ (99*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (c + I*d)*((2*(c + d*T
an[e + f*x])^(3/2))/3 + (c + I*d)*((2*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(-c - I
*d) + 2*Sqrt[c + d*Tan[e + f*x]]))))/f))/(9*d)))/(11*d)

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Maple [B]  time = 0.192, size = 11478, normalized size = 22.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(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((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(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((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^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((a+b*tan(f*x+e))**2*(c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Timed out

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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 )}^{2}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(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)^2*(d*tan(f*x + e) + c)^(5/2), x)