\(\int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx\) [662]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 89, 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, 926, 95, 211} \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}\right )}{\sqrt {2-3 i} d}+\frac {\arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}\right )}{\sqrt {2+3 i} d} \]

[In]

Int[1/(Sqrt[-3 - 2*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]

[Out]

ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTan[(Sqrt[2 + 3*I]
*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]]/(Sqrt[2 + 3*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 926

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

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 {1}{\sqrt {-3-2 x} \sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {i}{2 \sqrt {-3-2 x} (i-x) \sqrt {x}}+\frac {i}{2 \sqrt {-3-2 x} \sqrt {x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {i \text {Subst}\left (\int \frac {1}{\sqrt {-3-2 x} (i-x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {-3-2 x} \sqrt {x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {i \text {Subst}\left (\int \frac {1}{i-(3-2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{d}+\frac {i \text {Subst}\left (\int \frac {1}{i+(3+2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{d} \\ & = \frac {\arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {-\sqrt {2+3 i} \arctan \left (\frac {\sqrt {\frac {2}{13}+\frac {3 i}{13}} \sqrt {-3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-2+3 i} \text {arctanh}\left (\frac {\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {-3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \]

[In]

Integrate[1/(Sqrt[-3 - 2*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]

[Out]

(-(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[-3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]) + Sqrt[-2 + 3*I
]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[-3 - 2*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. \(438\) vs. \(2(73)=146\).

Time = 3.85 (sec) , antiderivative size = 439, normalized size of antiderivative = 4.93

method	result	size
derivativedivides	\(-\frac {\sqrt {-3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (4 \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-17 \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-18 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+36 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\)	\(439\)
default	\(-\frac {\sqrt {-3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (4 \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-17 \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-18 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+36 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\)	\(439\)

[In]

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

[Out]

-1/2/d*(-3-2*tan(d*x+c))^(1/2)*(-tan(d*x+c)*(3+2*tan(d*x+c))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2)*(13^(1/2)-2+3*
tan(d*x+c))*(4*13^(1/2)*(2*13^(1/2)+4)^(1/2)*arctanh(1/6318*(-4+2*13^(1/2))^(1/2)*(4*13^(1/2)+17)*(13^(1/2)+2-
3*tan(d*x+c))*(17*13^(1/2)-52)/(13^(1/2)-2+3*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(3+2*tan(d*x+c))/(13^(1/2)-2+3*
tan(d*x+c))^2)^(1/2))*(-4+2*13^(1/2))^(1/2)-17*(2*13^(1/2)+4)^(1/2)*arctanh(1/6318*(-4+2*13^(1/2))^(1/2)*(4*13
^(1/2)+17)*(13^(1/2)+2-3*tan(d*x+c))*(17*13^(1/2)-52)/(13^(1/2)-2+3*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(3+2*tan
(d*x+c))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2))*(-4+2*13^(1/2))^(1/2)-18*arctan(6*13^(1/2)*(-tan(d*x+c)*(3+2*tan(
d*x+c))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+52)^(1/2))*13^(1/2)+36*arctan(6*13^(1/2)*(-tan(d*x+c)*
(3+2*tan(d*x+c))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+52)^(1/2)))/tan(d*x+c)^(1/2)/(2*13^(1/2)+4)^(
1/2)/(3+2*tan(d*x+c))/(17*13^(1/2)-52)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1485 vs. \(2 (65) = 130\).

Time = 0.34 (sec) , antiderivative size = 1485, normalized size of antiderivative = 16.69 \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*sqrt(1/13)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(1/2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x
+ c) - (1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d)*sqrt(-(3*d^2*sqrt(-1/d
^4) + 2)/d^2) + 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) - 253*tan(d*x + c) - 204)*sqrt(-2*tan(d*x + c
) - 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(-1/2
*(sqrt(1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) - (1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 7
59*d^3)*sqrt(-1/d^4) + 612*d)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2) + 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-
1/d^4) - 253*tan(d*x + c) - 204)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqr
t(1/13)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(1/2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) -
(1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2
)/d^2) - 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) - 253*tan(d*x + c) - 204)*sqrt(-2*tan(d*x + c) - 3)*
sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(-1/2*(sqrt(
1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) - (1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)
*sqrt(-1/d^4) + 612*d)*sqrt(-(3*d^2*sqrt(-1/d^4) + 2)/d^2) - 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4)
- 253*tan(d*x + c) - 204)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)
*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2)*log(1/2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) + (1575*d^
3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2) +
2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) + 253*tan(d*x + c) + 204)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(
d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2)*log(-1/2*(sqrt(1/13)*(400
*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) + (1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d
^4) + 612*d)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2) + 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) + 253*tan(d
*x + c) + 204)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt((3*d^
2*sqrt(-1/d^4) - 2)/d^2)*log(1/2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 + 2334*d*tan(d*x + c) + (1575*d^3*tan(d*x +
c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2) - 2*((204*d^2
*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) + 253*tan(d*x + c) + 204)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(d*x + c)))/
(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2)*log(-1/2*(sqrt(1/13)*(400*d*tan(d*x
+ c)^2 + 2334*d*tan(d*x + c) + (1575*d^3*tan(d*x + c)^2 + 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d
)*sqrt((3*d^2*sqrt(-1/d^4) - 2)/d^2) - 2*((204*d^2*tan(d*x + c) - 253*d^2)*sqrt(-1/d^4) + 253*tan(d*x + c) + 2
04)*sqrt(-2*tan(d*x + c) - 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))

Sympy [F]

\[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 2 \tan {\left (c + d x \right )} - 3} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (65) = 130\).

Time = 0.38 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.64 \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\frac {\left (1904 i + 1536\right ) \, \sqrt {2} \log \left (3 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} + \left (24 i + 18\right ) \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 27\right )}{23377 \, d} - \frac {\left (1904 i - 1536\right ) \, \sqrt {2} \log \left (3 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} - \left (24 i - 18\right ) \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 27\right )}{23377 \, d} - \frac {\left (12720 i + 27456\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 2 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (4 i - 3\right ) \, \sqrt {13} - 8 i + 6}{2 \, {\left (\sqrt {13} \sqrt {\sqrt {13} + 2} + \left (3 i + 2\right ) \, \sqrt {\sqrt {13} + 2}\right )}}\right )}{23377 \, d \sqrt {\sqrt {13} + 2} {\left (\frac {3 i}{\sqrt {13} + 2} + 1\right )}} - \frac {\left (12720 i - 27456\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 2 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (4 i + 3\right ) \, \sqrt {13} + 8 i + 6}{2 \, {\left (\sqrt {13} \sqrt {\sqrt {13} + 2} - \left (3 i - 2\right ) \, \sqrt {\sqrt {13} + 2}\right )}}\right )}{23377 \, d \sqrt {\sqrt {13} + 2} {\left (-\frac {3 i}{\sqrt {13} + 2} + 1\right )}} \]

[In]

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

[Out]

-(1904/23377*I + 1536/23377)*sqrt(2)*log(3*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^4 + (24*I
+ 18)*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^2 + 27)/d - (1904/23377*I - 1536/23377)*sqrt(
2)*log(3*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^4 - (24*I - 18)*(sqrt(2)*sqrt(-tan(d*x + c)
) - sqrt(-2*tan(d*x + c) - 3))^2 + 27)/d - (12720/23377*I + 27456/23377)*sqrt(2)*arctan(1/2*(sqrt(13)*(sqrt(2)
*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^2 + 2*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) -
3))^2 - (4*I - 3)*sqrt(13) - 8*I + 6)/(sqrt(13)*sqrt(sqrt(13) + 2) + (3*I + 2)*sqrt(sqrt(13) + 2)))/(d*sqrt(sq
rt(13) + 2)*(3*I/(sqrt(13) + 2) + 1)) - (12720/23377*I - 27456/23377)*sqrt(2)*arctan(1/2*(sqrt(13)*(sqrt(2)*sq
rt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^2 + 2*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))
^2 + (4*I + 3)*sqrt(13) + 8*I + 6)/(sqrt(13)*sqrt(sqrt(13) + 2) - (3*I - 2)*sqrt(sqrt(13) + 2)))/(d*sqrt(sqrt(
13) + 2)*(-3*I/(sqrt(13) + 2) + 1))

Mupad [B] (verification not implemented)

Time = 7.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.46 \[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5900168033545907947438080+14160403280510179073851392{}\mathrm {i}\right )}{\sqrt {-2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {1770050410063772384231424-737521004193238493429760{}\mathrm {i}}{d}+\frac {{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (3540100820127544768462848-1475042008386476986859520{}\mathrm {i}\right )}{d\,\left (2\,\mathrm {tan}\left (c+d\,x\right )+3\right )}\right )}\right )\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5900168033545907947438080-14160403280510179073851392{}\mathrm {i}\right )}{\sqrt {-2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {1770050410063772384231424+737521004193238493429760{}\mathrm {i}}{d}+\frac {{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (3540100820127544768462848+1475042008386476986859520{}\mathrm {i}\right )}{d\,\left (2\,\mathrm {tan}\left (c+d\,x\right )+3\right )}\right )}\right )\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]

[In]

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

[Out]

atan((((- 1/26 + 3i/52)/d^2)^(1/2)*((2^(1/2)*3^(1/2)*1i)/2 - tan(c + d*x)^(1/2))*(5900168033545907947438080 -
14160403280510179073851392i))/((- 2*tan(c + d*x) - 3)^(1/2)*((1770050410063772384231424 + 73752100419323849342
9760i)/d + (((2^(1/2)*3^(1/2)*1i)/2 - tan(c + d*x)^(1/2))^2*(3540100820127544768462848 + 147504200838647698685
9520i))/(d*(2*tan(c + d*x) + 3)))))*((- 1/26 + 3i/52)/d^2)^(1/2)*2i - atan((((- 1/26 - 3i/52)/d^2)^(1/2)*((2^(
1/2)*3^(1/2)*1i)/2 - tan(c + d*x)^(1/2))*(5900168033545907947438080 + 14160403280510179073851392i))/((- 2*tan(
c + d*x) - 3)^(1/2)*((1770050410063772384231424 - 737521004193238493429760i)/d + (((2^(1/2)*3^(1/2)*1i)/2 - ta
n(c + d*x)^(1/2))^2*(3540100820127544768462848 - 1475042008386476986859520i))/(d*(2*tan(c + d*x) + 3)))))*((-
1/26 - 3i/52)/d^2)^(1/2)*2i