3.13.69 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx\) [1269]

3.13.69.1 Optimal result

Integrand size = 29, antiderivative size = 206 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{3/2} f}+\frac {i \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2} f}-\frac {2 b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}} \]

output

-I*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c+d*tan(f*x 
+e))^(1/2))*(c-I*d)^(1/2)/(a-I*b)^(3/2)/f+I*arctanh((c+I*d)^(1/2)*(a+b*tan 
(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))*(c+I*d)^(1/2)/(a+I*b) 
^(3/2)/f-2*b*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))^(1/2)
 

3.13.69.2 Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\frac {\frac {i \sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\frac {(i a+b) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {2 b \sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}}{a-i b}}{f} \]

input

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

output

((I*Sqrt[-c + I*d]*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)^(3/2) + (((I*a + b)*Sqrt 
[c + I*d]*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)^(3/2) - (2*b*Sqrt[c + d*Tan[e + f*x] 
])/((a + I*b)*Sqrt[a + b*Tan[e + f*x]]))/(a - I*b))/f
 

3.13.69.3 Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3042, 4051, 27, 3042, 4099, 3042, 4098, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

\(\displaystyle -\frac {2 \int -\frac {a c+b d-(b c-a d) \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a c+b d-(b c-a d) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a c+b d-(b c-a d) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\)

\(\Big \downarrow \) 4099

\(\displaystyle -\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {1}{2} (a-i b) (c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b) (c-i d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {1}{2} (a-i b) (c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b) (c-i d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 4098

\(\displaystyle -\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {(a+i b) (c-i d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {(a-i b) (c+i d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}}{a^2+b^2}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {(a-i b) (c+i d) \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {(a+i b) (c-i d) \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}}{a^2+b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {i (a-i b) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i (a+i b) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}}{a^2+b^2}\)

input

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

output

(((-I)*(a + I*b)*Sqrt[c - I*d]*ArcTanh[(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]*f) + (I*(a 
 - I*b)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sq 
rt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f))/(a^2 + b^2) - ( 
2*b*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])
 

3.13.69.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4051

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + 
d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[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] && Int 
egerQ[2*m]
 

rule 4098

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[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 4099

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(A + I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T 
an[e + f*x]), x], x] + Simp[(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]
 

3.13.69.4 Maple [F(-1)]

Timed out.

\[\int \frac {\sqrt {c +d \tan \left (f x +e \right )}}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.13.69.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11981 vs. \(2 (158) = 316\).

Time = 7.67 (sec) , antiderivative size = 11981, normalized size of antiderivative = 58.16 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

input

integrate((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^(3/2),x, algorithm="fric 
as")
 

output

Too large to include
 

3.13.69.6 Sympy [F]

\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

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

output

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

3.13.69.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^(3/2),x, algorithm="maxi 
ma")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((2*b*d+2*a*c)^2>0)', see `assum 
e?` for mo
 

3.13.69.8 Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

input

integrate((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^(3/2),x, algorithm="giac 
")
 

output

Timed out
 

3.13.69.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]

input

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

output

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