\(\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx\) [665]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]

[Out]

-I*arctan((3-2*I)^(1/2)*tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2))/d/(3-2*I)^(1/2)+I*arctan((3+2*I)^(1/2)*tan(d*
x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2))/d/(3+2*I)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3656, 924, 95, 211} \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d}-\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d} \]

[In]

Int[Sqrt[Tan[c + d*x]]/Sqrt[2 - 3*Tan[c + d*x]],x]

[Out]

((-I)*ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]])/(Sqrt[3 - 2*I]*d) + (I*ArcTan[(Sqrt
[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[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 {x}}{\sqrt {2-3 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{2 \sqrt {2-3 x} (i-x) \sqrt {x}}+\frac {1}{2 \sqrt {2-3 x} \sqrt {x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} (i-x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{i-(2-3 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{i+(2+3 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{d} \\ & = -\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\frac {i \left (\sqrt {3+2 i} \arctan \left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \text {arctanh}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \]

[In]

Integrate[Sqrt[Tan[c + d*x]]/Sqrt[2 - 3*Tan[c + d*x]],x]

[Out]

(I*(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]] + Sqrt[-3 + 2*I]
*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(77)=154\).

Time = 4.21 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.61

method	result	size
derivativedivides	\(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\)	\(438\)
default	\(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\)	\(438\)

[In]

int(tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/d*(2-3*tan(d*x+c))^(1/2)*(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2)*(13^(1/2)-3-2*t
an(d*x+c))*(arctanh(1/208*(2*13^(1/2)-6)^(1/2)*(3*13^(1/2)+11)*(13^(1/2)+3+2*tan(d*x+c))*(11*13^(1/2)-39)/(13^
(1/2)-3-2*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2))*(2*13^(1/2)-
6)^(1/2)*13^(1/2)*(2*13^(1/2)+6)^(1/2)-3*arctanh(1/208*(2*13^(1/2)-6)^(1/2)*(3*13^(1/2)+11)*(13^(1/2)+3+2*tan(
d*x+c))*(11*13^(1/2)-39)/(13^(1/2)-3-2*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2*tan(d
*x+c))^2)^(1/2))*(2*13^(1/2)-6)^(1/2)*(2*13^(1/2)+6)^(1/2)+12*arctan(4*13^(1/2)*(-tan(d*x+c)*(-2+3*tan(d*x+c))
/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+78)^(1/2))*13^(1/2)-44*arctan(4*13^(1/2)*(-tan(d*x+c)*(-2+3*t
an(d*x+c))/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+78)^(1/2)))/tan(d*x+c)^(1/2)/(-2+3*tan(d*x+c))/(2*1
3^(1/2)+6)^(1/2)/(11*13^(1/2)-39)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (67) = 134\).

Time = 0.35 (sec) , antiderivative size = 1477, normalized size of antiderivative = 15.55 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) +
(135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d
^2) + ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(
d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log(-(sqrt(1/13)*(155*d*t
an(d*x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 5
6*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2) + ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)
*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4)
+ 3)/d^2)*log((sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(
d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2) - ((33*d^2*tan(d*x + c) + 56*d^2)*s
qrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*
sqrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log(-(sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) + (1
35*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)
- ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x
+ c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2)*log((sqrt(1/13)*(155*d*tan(
d*x + c)^2 + 102*d*tan(d*x + c) - (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d
)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2) + ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) + 56*tan(d*x + c) - 33)*s
qrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4)
- 3)/d^2)*log(-(sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) - (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(
d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2) + ((33*d^2*tan(d*x + c) + 56*d^2)*
sqrt(-1/d^4) + 56*tan(d*x + c) - 33)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8
*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2)*log((sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) - (
135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^
2) - ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) + 56*tan(d*x + c) - 33)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d
*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2)*log(-(sqrt(1/13)*(155*d*t
an(d*x + c)^2 + 102*d*tan(d*x + c) - (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 5
6*d)*sqrt(-(2*d^2*sqrt(-1/d^4) - 3)/d^2) - ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) + 56*tan(d*x + c) - 33
)*sqrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))

Sympy [F]

\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {2 - 3 \tan {\left (c + d x \right )}}}\, dx \]

[In]

integrate(tan(d*x+c)**(1/2)/(2-3*tan(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(tan(c + d*x))/sqrt(2 - 3*tan(c + d*x)), x)

Maxima [F]

\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} \,d x } \]

[In]

integrate(tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(tan(d*x + c))/sqrt(-3*tan(d*x + c) + 2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (67) = 134\).

Time = 0.90 (sec) , antiderivative size = 641, normalized size of antiderivative = 6.75 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=-\frac {1}{2028} \, \sqrt {3} {\left (\frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} + \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} - \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} + \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} + 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}} - \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} - 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}}\right )} \]

[In]

integrate(tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/2028*sqrt(3)*(2*(2*d^2*sqrt(1014*sqrt(13) - 702) - 3*d*sqrt(1014*sqrt(13) + 702)*abs(d))*arctan(13/18*(9/13
)^(3/4)*(2*(9/13)^(1/4)*sqrt(-3/26*sqrt(13) + 1/2) + (sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x +
c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))/sqrt(3/26*sqrt(13) + 1/2))/d^3 + 2
*(2*d^2*sqrt(1014*sqrt(13) - 702) - 3*d*sqrt(1014*sqrt(13) + 702)*abs(d))*arctan(-13/18*(9/13)^(3/4)*(2*(9/13)
^(1/4)*sqrt(-3/26*sqrt(13) + 1/2) - (sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) + sqrt(-3
*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))/sqrt(3/26*sqrt(13) + 1/2))/d^3 + (2*d^2*sqrt(1014*s
qrt(13) + 702) + 3*d*sqrt(1014*sqrt(13) - 702)*abs(d))*log(((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan
(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))^2 + 4*(9/13)^(1/4)*sqrt(-3/
26*sqrt(13) + 1/2)*((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) +
2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))) + 12*sqrt(1/13))/d^3 - (2*d^2*sqrt(1014*sqrt(13) + 702) + 3*d*sqrt(
1014*sqrt(13) - 702)*abs(d))*log(((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*t
an(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))^2 - 4*(9/13)^(1/4)*sqrt(-3/26*sqrt(13) + 1/2)*((sqrt(
3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x +
c)) - sqrt(2))) + 12*sqrt(1/13))/d^3)

Mupad [B] (verification not implemented)

Time = 6.01 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6-4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6+4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]

[In]

int(tan(c + d*x)^(1/2)/(2 - 3*tan(c + d*x))^(1/2),x)

[Out]

atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 - 1i/26)/d^2)^(1/2)*(6 + 4i) - d*tan(c + d*x)^(1/2)*((3/52 - 1i/26)/
d^2)^(1/2)*(2 - 3*tan(c + d*x))^(1/2)*(6 + 4i))/(3*tan(c + d*x) + 2^(1/2)*(2 - 3*tan(c + d*x))^(1/2) - 2))*((3
/52 - 1i/26)/d^2)^(1/2)*2i - atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 + 1i/26)/d^2)^(1/2)*(6 - 4i) - d*tan(c
+ d*x)^(1/2)*((3/52 + 1i/26)/d^2)^(1/2)*(2 - 3*tan(c + d*x))^(1/2)*(6 - 4i))/(3*tan(c + d*x) + 2^(1/2)*(2 - 3*
tan(c + d*x))^(1/2) - 2))*((3/52 + 1i/26)/d^2)^(1/2)*2i