3.1240 \(\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=341 \[ \frac{\left (-2 a^2 b^2 \left (12 c^2-13 d^2\right )+24 a^3 b c d-3 a^4 d^2-40 a b^3 c d+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} f \left (a^2+b^2\right )^3 \sqrt{b c-a d}}-\frac{\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{(c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (b+i a)^3}+\frac{(c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-b+i a)^3} \]

[Out]

-(((c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(3/2)*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2
*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*
Sqrt[b]*(a^2 + b^2)^3*Sqrt[b*c - a*d]*f) - ((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[
e + f*x])^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]
))

________________________________________________________________________________________

Rubi [A]  time = 1.86867, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {3567, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{\left (-2 a^2 b^2 \left (12 c^2-13 d^2\right )+24 a^3 b c d-3 a^4 d^2-40 a b^3 c d+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} f \left (a^2+b^2\right )^3 \sqrt{b c-a d}}-\frac{\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{(c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (b+i a)^3}+\frac{(c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-b+i a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(3/2)*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2
*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*
Sqrt[b]*(a^2 + b^2)^3*Sqrt[b*c - a*d]*f) - ((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[
e + f*x])^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]
))

Rule 3567

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

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 3653

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]

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]

Rule 3634

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]

Rubi steps

\begin{align*} \int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx &=-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\int \frac{\frac{1}{2} \left (-5 b c d-a \left (4 c^2-d^2\right )\right )-2 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac{3}{2} d (b c-a d) \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\int \frac{\frac{1}{4} (b c-a d) \left (8 a^2 c^2-8 b^2 c^2+24 a b c d-5 a^2 d^2+3 b^2 d^2\right )-4 (b c-a d)^2 (a c+b d) \tan (e+f x)-\frac{1}{4} d (b c-a d) \left (8 a b c-3 a^2 d+5 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\int \frac{2 (b c-a d) \left (a^3 c^2-3 a b^2 c^2+6 a^2 b c d-2 b^3 c d-a^3 d^2+3 a b^2 d^2\right )-2 (b c-a d) \left (3 a^2 b c^2-b^3 c^2-2 a^3 c d+6 a b^2 c d-3 a^2 b d^2+b^3 d^2\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^3 (b c-a d)}-\frac{\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\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 \left (a^2+b^2\right )^3}\\ &=-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{(c-i d)^2 \int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac{(c+i d)^2 \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac{\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f}\\ &=-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{(c-i d)^2 \operatorname{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{(c+i d)^2 \operatorname{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 (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \operatorname{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 \left (a^2+b^2\right )^3 d f}\\ &=\frac{\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \left (a^2+b^2\right )^3 \sqrt{b c-a d} f}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{(c-i d)^2 \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 )}{(a-i b)^3 d f}-\frac{(c+i d)^2 \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 )}{(a+i b)^3 d f}\\ &=-\frac{(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{(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 (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \left (a^2+b^2\right )^3 \sqrt{b c-a d} f}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end{align*}

Mathematica [B]  time = 6.33771, size = 2093, normalized size = 6.14 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*(c + d*Tan[e + f*x])^(5/2))/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[e
+ f*x])^(3/2))/(f*(a + b*Tan[e + f*x]))) + (2*(-(b*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(2*f*(a + b*Tan[e +
f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b
*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d -
a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b
*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^
2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) - I*(a*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^
3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*(b*c
- a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d
)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8
- a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2)))*ArcTanh[Sqrt[c + d*Ta
n[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3
*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c
- a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)
^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 -
a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) + I*(a*(b*c - a*d)*((3*b
^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d
- b*d^2))/2) - b*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d
)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a
*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))
/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*(-(a*b
*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)
*(b*c^2 - 2*a*c*d - b*d^2))/2)) + (a^2*d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c
- a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2 + b^2*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d -
a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b
*c^2 - 2*a*c*d - b*d^2))/2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2
)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*
b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2
)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b))/b)/(2*(a^2 + b^2)*(b*c - a*d))

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Maple [B]  time = 0.094, size = 4733, normalized size = 13.9 \begin{align*} \text{output too large to display} \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))^3,x)

[Out]

2/f*d/(a^2+b^2)^3/(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^3*c+3/4/f*d/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2-2/f*d/(a^2+b^2)^3/(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^3*
c+6/f*d/(a^2+b^2)^3/(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*b^2*c-3/f*d/(a^2+b^2)^3/(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^2+d^2)^(1/2)*a*b^2+2/f*d/(a^2+b
^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a*b^4*c^2-6/f*d/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+
d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^3*b*c+1/4/f/d/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*c-2/f*d/(a^
2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(3/2)*a^3*c*b^2+3/f/(a^2+b^2)^3/(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^2+d^2)^(1/2
)*a^2*b*c-3/f/(a^2+b^2)^3/(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^2+d^2)^(1/2)*a^2*b*c-2/f*d/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d
*tan(f*x+e))^(3/2)*a*c*b^4-13/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^4*b*c+2/f*d/
(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^3*b^2*c^2-5/2/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*
d)^2*(c+d*tan(f*x+e))^(1/2)*a^2*b^3*c+3/4/f/d/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2+10/f*d/(a^2+b^2)^3/((a*d-b*c)*b)^(
1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a*b^3*c-6/f*d/(a^2+b^2)^3/(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*b^2*c+3/f
*d/(a^2+b^2)^3/(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^2+d^2)^(1/2)*a*b^2-1/4/f/d/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*c-3/4/f/d/
(a^2+b^2)^3*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*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*b^2*c^2+1/f*d^2/(a^2+b^2)^3/(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^3-2/f/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arcta
n((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^4*c^2+1/4/f/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3-1/2/f/(a
^2+b^2)^3*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*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b^3*c-1/f/(a^2+b^2)^3/(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^3*c^2-1/4/f/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3+1
/f/(a^2+b^2)^3/(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^3*c^2+1/2/f/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c-5/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*
d)^2*(c+d*tan(f*x+e))^(3/2)*b^5+5/4/f*d^3/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^5+3/4/f*
d^2/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^4+3/4/f*d^2/(a^2+b^
2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^4-1/4/f*d/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a^3-1/f*d^2/(a^2+b^2)^3/(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^3+1/4/f*d/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3-1/4/f/d/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2
-3/f/(a^2+b^2)^3/(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^2*b*c^2-3/2/f/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+6/f/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2
)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2*b^2*c^2+3/4/f/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a
^2*b-1/f/(a^2+b^2)^3/(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^2+d^2)^(1/2)*b^3*c+1/f/(a^2+b^2)^3/(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^2+d^2)^(1/2)*b^3*c
+3/f/(a^2+b^2)^3/(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^2*b*c^2+1/f*d/(a^2+b^2)^3/(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^2+d^2)^(1/2)*a^3-3/f*d^2/(a^2+b^
2)^3/(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^2*b+3/f*d^2/(a^2+b^2)^3/(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^2*b-3/4/f/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*ta
n(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a
^2*b+3/2/f/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+3/4/f/d/(a^2+b^2)^3*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*c-3/4/f/d/(a^2+b^2)^3
*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))*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*c-3/4/f*d/(a^2+b^2)^3*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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2+3/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^
2*(c+d*tan(f*x+e))^(3/2)*a^4*b-1/2/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(3/2)*a^2*b^3+1/2
/f*d^3/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^3*b^2-3/4/f*d^3/(a^2+b^2)^3/(tan(f*x+e)*b*d
+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a*b^4+3/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*b^5*c
-13/2/f*d^2/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2*b^2+1/4/f
/d/(a^2+b^2)^3*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*(c^2
+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2-1/f*d/(a^2+b^2)^3/(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^2+d^2)^(1/2)*a^3

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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))^3,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))^3,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))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \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))^3,x, algorithm="giac")

[Out]

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