\(\int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+b \tan (e+f x)}} \, dx\) [1268]

Optimal result

Integrand size = 29, antiderivative size = 163 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+b \tan (e+f x)}} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b} f}+\frac {i \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} f} \]

[Out]

-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)/f/(a-I*b)^
(1/2)+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)/f/(a+
I*b)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3656, 924, 95, 214} \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+b \tan (e+f x)}} \, dx=\frac {i \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}} \]

[In]

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

[Out]

((-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]])]
)/(Sqrt[a - I*b]*f) + (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]])])/(Sqrt[a + I*b]*f)

Rule 95

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 214

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 924

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegr
and[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &
& NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]

Rule 3656

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {c+d x}}{\sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {i c-d}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {i c+d}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i c-d) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {(i c+d) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {(i c-d) \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {(i c+d) \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{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 )}{\sqrt {a-i b} 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 )}{\sqrt {a+i b} f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+b \tan (e+f x)}} \, dx=-\frac {i \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+i b}}-\frac {\sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b}}\right )}{f} \]

[In]

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

[Out]

((-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]])])/Sqrt[-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

Maple [F(-1)]

Timed out.

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6555 vs. \(2 (119) = 238\).

Time = 3.69 (sec) , antiderivative size = 6555, normalized size of antiderivative = 40.21 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+b \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}