3.1234 \(\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=342 \[ \frac{\sqrt{b} \left (-6 a^2 b^2 \left (4 c^2-3 d^2\right )+40 a^3 b c d-15 a^4 d^2-24 a b^3 c d+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 f \left (a^2+b^2\right )^3 (b c-a d)^{3/2}}-\frac{b \left (-7 a^2 d+8 a b c+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (b c-a d) (a+b \tan (e+f x))}-\frac{b \sqrt{c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{\sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (b+i a)^3}+\frac{\sqrt{c+i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-b+i a)^3} \]

[Out]

-((Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + (Sqrt[c + I*d]*ArcTanh[Sq
rt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b]*(40*a^3*b*c*d - 24*a*b^3*c*d - 15*a^4*d^2 -
6*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(8*c^2 + d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4
*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - (b*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) -
(b*(8*a*b*c - 7*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.51729, antiderivative size = 342, 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 = {3568, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{\sqrt{b} \left (-6 a^2 b^2 \left (4 c^2-3 d^2\right )+40 a^3 b c d-15 a^4 d^2-24 a b^3 c d+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 f \left (a^2+b^2\right )^3 (b c-a d)^{3/2}}-\frac{b \left (-7 a^2 d+8 a b c+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (b c-a d) (a+b \tan (e+f x))}-\frac{b \sqrt{c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{\sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (b+i a)^3}+\frac{\sqrt{c+i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-b+i a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + (Sqrt[c + I*d]*ArcTanh[Sq
rt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b]*(40*a^3*b*c*d - 24*a*b^3*c*d - 15*a^4*d^2 -
6*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(8*c^2 + d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4
*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - (b*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) -
(b*(8*a*b*c - 7*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x]))

Rule 3568

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(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 - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*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] && GtQ[n, 0] && 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{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^3} \, dx &=-\frac{b \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} (-4 a c-b d)+2 (b c-a d) \tan (e+f x)+\frac{3}{2} b 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 \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac{\int \frac{\frac{1}{4} \left (-8 a^3 c d+16 a b^2 c d+4 a^2 b \left (2 c^2-\frac{9 d^2}{4}\right )-b^3 \left (8 c^2+d^2\right )\right )-2 (b c-a d) \left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)-\frac{1}{4} b d \left (8 a b c-7 a^2 d+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 \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac{\int \frac{2 (b c-a d) \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )-2 (b c-a d) \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\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 (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+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 \left (a^2+b^2\right )^3 (b c-a d)}\\ &=-\frac{b \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac{(c-i d) \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) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac{\left (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\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 (b c-a d) f}\\ &=-\frac{b \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}-\frac{(c+i d) \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{(i c+d) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^3 f}-\frac{\left (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\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 (b c-a d) f}\\ &=\frac{\sqrt{b} \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac{b \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}-\frac{(c+i d) \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{(i c+d) \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 )}{(i a+b)^3 d f}\\ &=-\frac{\sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{(i a+b)^3 f}+\frac{\sqrt{c+i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{(i a-b)^3 f}+\frac{\sqrt{b} \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{4 \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac{b \sqrt{c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}\\ \end{align*}

Mathematica [B]  time = 6.35292, size = 747, normalized size = 2.18 \[ -\frac{b^2 (c+d \tan (e+f x))^{3/2}}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}-\frac{-\frac{b d \sqrt{c+d \tan (e+f x)}}{f (a+b \tan (e+f x))}-\frac{2 \left (-\frac{\left (\frac{1}{4} b^3 (b c-a d) (4 a c+b d)-a \left (\frac{3}{4} a b^2 d (b c-a d)-b^2 (b c-a d)^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac{\frac{2 \sqrt{b c-a d} \left (-\frac{1}{8} b^3 (b c-a d) \left (8 a^2 b c^2-9 a^2 b d^2-8 a^3 c d+16 a b^2 c d-8 b^3 c^2-b^3 d^2\right )+\frac{1}{8} a^2 b^2 d (b c-a d) \left (-7 a^2 d+8 a b c+b^2 d\right )-a b^2 (b c-a d)^2 \left (a^2 (-d)+2 a b c+b^2 d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\sqrt{b} f \left (a^2+b^2\right ) (a d-b c)}+\frac{\frac{i \sqrt{c-i d} \left (-b \left (3 a^2 b d+a^3 c-3 a b^2 c-b^3 d\right ) (b c-a d)^2-i b \left (3 a^2 b c+a^3 (-d)+3 a b^2 d-b^3 c\right ) (b c-a d)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (-c+i d)}-\frac{i \sqrt{c+i d} \left (-b (b c-a d)^2 \left (3 a^2 b d+a^3 c-3 a b^2 c-b^3 d\right )+i b (b c-a d)^2 \left (3 a^2 b c+a^3 (-d)+3 a b^2 d-b^3 c\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-c-i d)}}{a^2+b^2}}{\left (a^2+b^2\right ) (b c-a d)}\right )}{b}}{2 \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*(c + d*Tan[e + f*x])^(3/2))/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*Sqrt[c + d*Ta
n[e + f*x]])/(f*(a + b*Tan[e + f*x]))) - (2*(-((((I*Sqrt[c - I*d]*((-I)*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c - a
^3*d + 3*a*b^2*d) - b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/
Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(I*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) -
b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*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^2*b^2*d*(b*c - a*d)*(8*a*b*c - 7*a^2*d + b^2*d))/8 - a*b^2*(
b*c - a*d)^2*(2*a*b*c - a^2*d + b^2*d) - (b^3*(b*c - a*d)*(8*a^2*b*c^2 - 8*b^3*c^2 - 8*a^3*c*d + 16*a*b^2*c*d
- 9*a^2*b*d^2 - b^3*d^2))/8)*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^3*(b*c - a*d)*(4*a*c + b*d))/4 - a*((3*a*b^2*d*(b*c - a*
d))/4 - b^2*(b*c - a*d)^2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b)/(2
*(a^2 + b^2)*(b*c - a*d))

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Maple [B]  time = 0.109, size = 3203, normalized size = 9.4 \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))^(1/2)/(a+b*tan(f*x+e))^3,x)

[Out]

-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+1/4/f*d^2*b^5/(a^2+b^2)^3/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x
+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))-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))*a*b^2+2/f*d*b^3/(a^2+b^2)^3/(tan(f*x+e)*b*d
+a*d)^2/(a*d-b*c)*(c+d*tan(f*x+e))^(3/2)*a^3*c-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)*b^3+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)*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))*(c^2+d^2)^(1/2)*b^3-3/4/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)*a^2*b+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^3-1/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+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+3/4/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)*a^2*b+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))*a^3-1/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))*a^3-1/4/f*d^2*b^5/(a^2+
b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)+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))*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+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+1/4/f*d^2*b^6/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)
^2/(a*d-b*c)*(c+d*tan(f*x+e))^(3/2)+2/f*b^5/(a^2+b^2)^3/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^
(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2-3/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+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+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-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-9/4
/f*d^2*b/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^4-5/2/f*d^2*b^3/(a^2+b^2)^3/(tan(f*x+e)*b
*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^2+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+2/f*d*b^5/(a^2+b^2)^3/(ta
n(f*x+e)*b*d+a*d)^2/(a*d-b*c)*(c+d*tan(f*x+e))^(3/2)*a*c+10/f*d*b^2/(a^2+b^2)^3/(a*d-b*c)/((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*c-6/f*d*b^4/(a^2+b^2)^3/(a*d-b*c)/((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*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+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-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+9/2/f*d^2*b^3/(a^2+b^2)^3/
(a*d-b*c)/((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^2-3/2/f*d^2*b^4/(a^2+b^2)
^3/(tan(f*x+e)*b*d+a*d)^2/(a*d-b*c)*(c+d*tan(f*x+e))^(3/2)*a^2+2/f*d*b^4/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c
+d*tan(f*x+e))^(1/2)*a*c-15/4/f*d^2*b/(a^2+b^2)^3/(a*d-b*c)/((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^4-7/4/f*d^2*b^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2/(a*d-b*c)*(c+d*tan(f*x+e))^(3/2)*a
^4-3/4/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/f*d*b^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan(f*x+e))^(1/2)*a^3
*c-6/f*b^3/(a^2+b^2)^3/(a*d-b*c)/((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*
c^2

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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))^(1/2)/(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))^(1/2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)/(a+b*tan(f*x+e))**3,x)

[Out]

Integral(sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \tan \left (f x + e\right ) + c}}{{\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))^(1/2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

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