3.1303 \(\int \frac{1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=596 \[ -\frac{4 d \left (-a^2 b^3 d \left (28 c^2 d^2+7 c^4+15 d^4\right )+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a^5 c d^4+3 a b^4 c \left (2 c^2 d^2+c^4+2 d^4\right )-b^5 d \left (15 c^2 d^2+4 c^4+8 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2 (b c-a d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{2 d \left (a^2 b^2 d \left (13 c^2+15 d^2\right )+a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (-2 a^2 d+a b c-b^2 d\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2} (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2} (c+i d)^{5/2}} \]

[Out]

((-I)*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)^(
5/2)*(c - I*d)^(5/2)*f) + (I*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)^(5/2)*(c + I*d)^(5/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/
2)*(c + d*Tan[e + f*x])^(3/2)) - (4*b^2*(a*b*c - 2*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*T
an[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (2*d*(a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(7*c^2 + 8*d^2) + a^2
*b^2*d*(13*c^2 + 15*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e
+ f*x])^(3/2)) - (4*d*(3*a^5*c*d^4 + 6*a^3*b^2*c*d^4 - a^4*b*d^3*(7*c^2 + 4*d^2) + 3*a*b^4*c*(c^4 + 2*c^2*d^2
+ 2*d^4) - b^5*d*(4*c^4 + 15*c^2*d^2 + 8*d^4) - a^2*b^3*d*(7*c^4 + 28*c^2*d^2 + 15*d^4))*Sqrt[a + b*Tan[e + f*
x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^4*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])

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Rubi [A]  time = 2.99694, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {3569, 3649, 3616, 3615, 93, 208} \[ -\frac{4 d \left (-a^2 b^3 d \left (28 c^2 d^2+7 c^4+15 d^4\right )+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a^5 c d^4+3 a b^4 c \left (2 c^2 d^2+c^4+2 d^4\right )-b^5 d \left (15 c^2 d^2+4 c^4+8 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2 (b c-a d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{2 d \left (a^2 b^2 d \left (13 c^2+15 d^2\right )+a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (-2 a^2 d+a b c-b^2 d\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2} (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2} (c+i d)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-I)*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)^(
5/2)*(c - I*d)^(5/2)*f) + (I*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)^(5/2)*(c + I*d)^(5/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/
2)*(c + d*Tan[e + f*x])^(3/2)) - (4*b^2*(a*b*c - 2*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*T
an[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (2*d*(a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(7*c^2 + 8*d^2) + a^2
*b^2*d*(13*c^2 + 15*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e
+ f*x])^(3/2)) - (4*d*(3*a^5*c*d^4 + 6*a^3*b^2*c*d^4 - a^4*b*d^3*(7*c^2 + 4*d^2) + 3*a*b^4*c*(c^4 + 2*c^2*d^2
+ 2*d^4) - b^5*d*(4*c^4 + 15*c^2*d^2 + 8*d^4) - a^2*b^3*d*(7*c^4 + 28*c^2*d^2 + 15*d^4))*Sqrt[a + b*Tan[e + f*
x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^4*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

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{1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{2 \int \frac{-\frac{3}{2} \left (a b c-a^2 d-2 b^2 d\right )+\frac{3}{2} b (b c-a d) \tan (e+f x)+3 b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{4 \int \frac{-\frac{3}{4} \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )-\frac{3}{2} a b (b c-a d)^2 \tan (e+f x)-6 b^2 d \left (a b c-2 a^2 d-b^2 d\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^2}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{8 \int \frac{-\frac{3}{8} \left (3 a^5 c d^3+3 a b^4 c d \left (3 c^2+4 d^2\right )+3 a^3 b^2 c d \left (3 c^2+5 d^2\right )-a^4 b d^2 \left (9 c^2+8 d^2\right )+b^5 \left (3 c^4-14 c^2 d^2-16 d^4\right )-a^2 b^3 \left (3 c^4+35 c^2 d^2+30 d^4\right )\right )-\frac{9}{8} (b c-a d)^3 \left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)+\frac{3}{4} b d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{9 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right )}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{16 \int \frac{\frac{9}{16} (b c-a d)^4 (a c-b c-a d-b d) (a c+b c+a d-b d)-\frac{9}{8} (b c-a d)^4 (b c+a d) (a c-b d) \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{9 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{\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)^2 (c-i d)^2}+\frac{\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)^2 (c+i d)^2}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{\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)^2 (c-i d)^2 f}+\frac{\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)^2 (c+i d)^2 f}\\ &=-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{\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)^2 (c-i d)^2 f}+\frac{\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)^2 (c+i d)^2 f}\\ &=-\frac{i \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)^{5/2} (c-i d)^{5/2} f}+\frac{i \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)^{5/2} (c+i d)^{5/2} f}-\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.5589, size = 1050, normalized size = 1.76 \[ -\frac{2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{2 \left (-\frac{2 \left (-\frac{3}{2} \left (-d a^2+b c a-2 b^2 d\right ) b^2-a \left (\frac{3}{2} b^2 (b c-a d)-3 a b^2 d\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 \left (-\frac{2 \sqrt{a+b \tan (e+f x)} \left (-\frac{3}{4} \left (-d^2 a^4+2 b c d a^3-b^2 \left (c^2+15 d^2\right ) a^2+6 b^3 c d a+b^4 \left (c^2-8 d^2\right )\right ) d^2-c \left (6 b^2 c d \left (-2 d a^2+b c a-b^2 d\right )-\frac{3}{2} a b d (b c-a d)^2\right )\right )}{3 (a d-b c) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac{2 \left (-\frac{9 i \left (\frac{(a-i b)^2 \tan ^{-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 ) (c-i d)^2}{\sqrt{a+i b} \sqrt{-c-i d}}+\frac{(a+i b)^2 (c+i d)^2 \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{i b-a} \sqrt{c-i d}}\right ) (b c-a d)^4}{8 (a d-b c) \left (c^2+d^2\right ) f}-\frac{2 \left (d^2 \left (\left (\frac{b c}{2}-\frac{3 a d}{2}\right ) \left (6 b^2 c d \left (-2 d a^2+b c a-b^2 d\right )-\frac{3}{2} a b d (b c-a d)^2\right )-\frac{3}{4} \left (b d^2-\frac{3}{2} c (a d-b c)\right ) \left (-d^2 a^4+2 b c d a^3-b^2 \left (c^2+15 d^2\right ) a^2+6 b^3 c d a+b^4 \left (c^2-8 d^2\right )\right )\right )-c \left (\frac{3}{2} d (a d-b c) \left (6 b^2 \left (-2 d a^2+b c a-b^2 d\right ) d^2-\frac{3}{4} \left (-d^2 a^4+2 b c d a^3-b^2 \left (c^2+15 d^2\right ) a^2+6 b^3 c d a+b^4 \left (c^2-8 d^2\right )\right ) d+\frac{3}{2} a b c (b c-a d)^2\right )-b c \left (-\frac{3}{4} \left (-d^2 a^4+2 b c d a^3-b^2 \left (c^2+15 d^2\right ) a^2+6 b^3 c d a+b^4 \left (c^2-8 d^2\right )\right ) d^2-c \left (6 b^2 c d \left (-2 d a^2+b c a-b^2 d\right )-\frac{3}{2} a b d (b c-a d)^2\right )\right )\right )\right ) \sqrt{a+b \tan (e+f x)}}{(a d-b c) \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}\right )}{3 (a d-b c) \left (c^2+d^2\right )}\right )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

(-2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*((-3*b^
2*(a*b*c - a^2*d - 2*b^2*d))/2 - a*(-3*a*b^2*d + (3*b^2*(b*c - a*d))/2)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a +
b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*(-(c*((-3*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*
a^2*d - b^2*d))) - (3*d^2*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/
4)*Sqrt[a + b*Tan[e + f*x]])/(3*(-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*((((-9*I)/8)*(b*
c - a*d)^4*(((a - I*b)^2*(c - I*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]) + ((a + I*b)^2*(c + I*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])))/((-(b*c) + a*d)*(c
^2 + d^2)*f) - (2*(d^2*(((b*c)/2 - (3*a*d)/2)*((-3*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*a^2*d - b^2*d
)) - (3*(b*d^2 - (3*c*(-(b*c) + a*d))/2)*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c
^2 + 15*d^2)))/4) - c*((3*d*(-(b*c) + a*d)*((3*a*b*c*(b*c - a*d)^2)/2 + 6*b^2*d^2*(a*b*c - 2*a^2*d - b^2*d) -
(3*d*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4))/2 - b*c*(-(c*((-3
*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*a^2*d - b^2*d))) - (3*d^2*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2
+ b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4)))*Sqrt[a + b*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f*Sq
rt[c + d*Tan[e + f*x]])))/(3*(-(b*c) + a*d)*(c^2 + d^2))))/((a^2 + b^2)*(b*c - a*d))))/(3*(a^2 + b^2)*(b*c - a
*d))

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x)

[Out]

int(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),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(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),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(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),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(1/(a+b*tan(f*x+e))**(5/2)/(c+d*tan(f*x+e))**(5/2),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(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError