3.7.59 \(\int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {3+2 \tan (c+d x)}} \, dx\) [659]

Optimal. Leaf size=89 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]

[Out]

arctanh((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)+arctanh((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)

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Rubi [A]
time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3656, 926, 95, 214} \begin {gather*} \frac {\tanh ^{-1}\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 {\tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)+3}}\right )}{\sqrt {2+3 i} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTanh[(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 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 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*} \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {3+2 \tan (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {3+2 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {i}{2 (i-x) \sqrt {x} \sqrt {3+2 x}}+\frac {i}{2 \sqrt {x} (i+x) \sqrt {3+2 x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {3+2 x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {3+2 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 {\tanh ^{-1}\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 {\tanh ^{-1}\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*}

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Mathematica [A]
time = 0.19, size = 89, normalized size = 1.00 \begin {gather*} \frac {\text {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 {\tanh ^{-1}\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 {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(479\) vs. \(2(73)=146\).
time = 1.17, size = 480, normalized size = 5.39

method	result	size
derivativedivides	\(\frac {\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}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \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 ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-17 \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \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 ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}+18 \arctanh \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 \arctanh \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 {3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\)	\(480\)
default	\(\frac {\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}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \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 ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-17 \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \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 ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}+18 \arctanh \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 \arctanh \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 {3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\)	\(480\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*(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)*arctan(1/56862*((17*13^(1/2)-52)*tan(d*x+c)*(3+2*tan(d*x+c))*(52+17*13^(1/2))/(13^(1/2)-2+3
*tan(d*x+c))^2)^(1/2)*(-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))/tan(d*x+c)/(3+2*tan(d*x+c)))*(-4+2*13^(1/2))^(1/2)-17*(2*13^(1/2)+4)^(1/2)*arctan(1/56862*(
(17*13^(1/2)-52)*tan(d*x+c)*(3+2*tan(d*x+c))*(52+17*13^(1/2))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2)*(-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))/tan(d*x+c)/(3+2*
tan(d*x+c)))*(-4+2*13^(1/2))^(1/2)+18*arctanh(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*arctanh(6*13^(1/2)*(tan(d*x+c)*(3+2*tan(d*x+c))/(13^(1/2)-2+3*t
an(d*x+c))^2)^(1/2)/(26*13^(1/2)+52)^(1/2)))/tan(d*x+c)^(1/2)/(3+2*tan(d*x+c))^(1/2)/(2*13^(1/2)+4)^(1/2)/(17*
13^(1/2)-52)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 \tan {\left (c + d x \right )} + 3} \sqrt {\tan {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (65) = 130\).
time = 0.53, size = 479, normalized size = 5.38 \begin {gather*} \frac {\sqrt {2} {\left (\sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} + 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} + \left (1647 i - 6954\right ) \, \sqrt {13} - \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} + 4266 i - 18012\right ) - \sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} - 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} + \left (1647 i - 6954\right ) \, \sqrt {13} + \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} + 4266 i - 18012\right ) - \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} + 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} - \left (450 i - 225\right ) \, \sqrt {13} - \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} + 540 i - 270\right ) + \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) + 3}\right )}^{2} - 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} - \left (450 i - 225\right ) \, \sqrt {13} + \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} + 540 i - 270\right )\right )}}{52 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/52*sqrt(2)*(sqrt(sqrt(13) - 2)*((9*I - 6)/(sqrt(13) - 2) - 2*I - 3)*log((915*I + 1098)*sqrt(13)*(sqrt(2)*sqr
t(tan(d*x + c)) - sqrt(2*tan(d*x + c) + 3))^2 + (2370*I + 2844)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x +
c) + 3))^2 + 366*sqrt(13)*sqrt(61*sqrt(13) + 158) + (1647*I - 6954)*sqrt(13) - (918*I - 948)*sqrt(61*sqrt(13)
+ 158) + 4266*I - 18012) - sqrt(sqrt(13) - 2)*((9*I - 6)/(sqrt(13) - 2) - 2*I - 3)*log((915*I + 1098)*sqrt(13
)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) + 3))^2 + (2370*I + 2844)*(sqrt(2)*sqrt(tan(d*x + c)) - sq
rt(2*tan(d*x + c) + 3))^2 - 366*sqrt(13)*sqrt(61*sqrt(13) + 158) + (1647*I - 6954)*sqrt(13) + (918*I - 948)*sq
rt(61*sqrt(13) + 158) + 4266*I - 18012) - sqrt(sqrt(13) + 2)*((6*I - 9)/(sqrt(13) + 2) - 3*I - 2)*log((90*I +
45)*sqrt(13)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) + 3))^2 - (108*I + 54)*(sqrt(2)*sqrt(tan(d*x +
c)) - sqrt(2*tan(d*x + c) + 3))^2 + 90*sqrt(13)*sqrt(5*sqrt(13) - 6) - (450*I - 225)*sqrt(13) - (306*I + 108)*
sqrt(5*sqrt(13) - 6) + 540*I - 270) + sqrt(sqrt(13) + 2)*((6*I - 9)/(sqrt(13) + 2) - 3*I - 2)*log((90*I + 45)*
sqrt(13)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) + 3))^2 - (108*I + 54)*(sqrt(2)*sqrt(tan(d*x + c))
- sqrt(2*tan(d*x + c) + 3))^2 - 90*sqrt(13)*sqrt(5*sqrt(13) - 6) - (450*I - 225)*sqrt(13) + (306*I + 108)*sqrt
(5*sqrt(13) - 6) + 540*I - 270))/d

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Mupad [B]
time = 6.44, size = 204, normalized size = 2.29 \begin {gather*} \mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}\,\left (-6-4{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {3}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}+3}\right )\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}\,\left (-6+4{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {6\,\mathrm {tan}\left (c+d\,x\right )+9}+3}\right )\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((3^(1/2)*d*tan(c + d*x)^(1/2)*((1/26 + 3i/52)/d^2)^(1/2)*(6 + 4i) - d*tan(c + d*x)^(1/2)*((1/26 + 3i/52)/
d^2)^(1/2)*(2*tan(c + d*x) + 3)^(1/2)*(6 + 4i))/(2*tan(c + d*x) - 3^(1/2)*(2*tan(c + d*x) + 3)^(1/2) + 3))*((1
/26 + 3i/52)/d^2)^(1/2)*2i - atan((3^(1/2)*d*tan(c + d*x)^(1/2)*((1/26 - 3i/52)/d^2)^(1/2)*(6 - 4i) - d*tan(c
+ d*x)^(1/2)*((1/26 - 3i/52)/d^2)^(1/2)*(2*tan(c + d*x) + 3)^(1/2)*(6 - 4i))/(2*tan(c + d*x) - (6*tan(c + d*x)
+ 9)^(1/2) + 3))*((1/26 - 3i/52)/d^2)^(1/2)*2i

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