Integrand size = 25, antiderivative size = 89 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {\arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \] Output:
arctan((3-2*I)^(1/2)*tan(d*x+c)^(1/2)/(-2-3*tan(d*x+c))^(1/2))/(3-2*I)^(1/ 2)/d+arctan((3+2*I)^(1/2)*tan(d*x+c)^(1/2)/(-2-3*tan(d*x+c))^(1/2))/(3+2*I )^(1/2)/d
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {-\sqrt {3+2 i} \arctan \left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \text {arctanh}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \] Input:
Integrate[1/(Sqrt[-2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]
Output:
(-(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)
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4058, 615, 2009}
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.
\(\displaystyle \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4058 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \frac {\int \left (\frac {i}{2 \sqrt {-3 \tan (c+d x)-2} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}+\frac {i}{2 \sqrt {-3 \tan (c+d x)-2} (\tan (c+d x)+i) \sqrt {\tan (c+d x)}}\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i}}+\frac {\arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i}}}{d}\) |
Input:
Int[1/(Sqrt[-2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]
Output:
(ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]]/Sqrt [3 - 2*I] + ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]]/Sqrt[3 + 2*I])/d
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(73)=146\).
Time = 0.58 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.80
method | result | size |
derivativedivides | \(\frac {\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (3 \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right ) \sqrt {\tan \left (d x +c \right )}}\) | \(427\) |
default | \(\frac {\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (3 \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right ) \sqrt {\tan \left (d x +c \right )}}\) | \(427\) |
Input:
int(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/d*(13^(1/2)-3+2*tan(d*x+c))*(3*13^(1/2)*(2*13^(1/2)+6)^(1/2)*(-6+2*13^ (1/2))^(1/2)*arctanh(1/208*(-6+2*13^(1/2))^(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))-11*(2*13^(1/2 )+6)^(1/2)*(-6+2*13^(1/2))^(1/2)*arctanh(1/208*(-6+2*13^(1/2))^(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))-4*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)+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)))/(-2-3*tan(d*x+c))^(1/2)/(2*13^(1/2)+6)^(1/2)/(11*13^(1/2)-39 )*(-tan(d*x+c)*(2+3*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^2)^(1/2)/tan(d*x +c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (65) = 130\).
Time = 0.16 (sec) , antiderivative size = 1477, normalized size of antiderivative = 16.60 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="fricas" )
Output:
1/8*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(135*d* tan(d*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*ta n(d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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)*(135*d*ta n(d*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan( d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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*s qrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(135*d*tan(d *x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan(d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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)*(135*d*tan(d*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan(d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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*sqr...
\[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 3 \tan {\left (c + d x \right )} - 2} \sqrt {\tan {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/(-2-3*tan(d*x+c))**(1/2)/tan(d*x+c)**(1/2),x)
Output:
Integral(1/(sqrt(-3*tan(c + d*x) - 2)*sqrt(tan(c + d*x))), x)
Exception generated. \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="maxima" )
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (65) = 130\).
Time = 0.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.02 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} - \left (6 i - 2\right ) \, \sqrt {13} - 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} + \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (\frac {2 i}{\sqrt {13} + 3} + 1\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + \left (6 i + 2\right ) \, \sqrt {13} + 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} - \left (2 i - 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )}} \] Input:
integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="giac")
Output:
-2*sqrt(3)*arctan((sqrt(13)*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x + c) - 2))^2 + 3*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x + c) - 2) )^2 - (6*I - 2)*sqrt(13) - 18*I + 6)/(sqrt(13)*sqrt(6*sqrt(13) + 18) + (2* I + 3)*sqrt(6*sqrt(13) + 18)))/(d*sqrt(6*sqrt(13) + 18)*(2*I/(sqrt(13) + 3 ) + 1)) + 2*sqrt(3)*arctan((sqrt(13)*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(- 3*tan(d*x + c) - 2))^2 + 3*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x + c) - 2))^2 + (6*I + 2)*sqrt(13) + 18*I + 6)/(sqrt(13)*sqrt(6*sqrt(13) + 18) - (2*I - 3)*sqrt(6*sqrt(13) + 18)))/(d*sqrt(6*sqrt(13) + 18)*(-2*I/(sq rt(13) + 3) + 1))
Time = 3.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}-2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}} \] Input:
int(1/(tan(c + d*x)^(1/2)*(- 3*tan(c + d*x) - 2)^(1/2)),x)
Output:
- 2*atanh((4*d*((- 3/52 - 1i/26)/d^2)^(1/2) + 6*d*tan(c + d*x)*((- 3/52 - 1i/26)/d^2)^(1/2))/(tan(c + d*x)^(1/2)*(- 3*tan(c + d*x) - 2)^(1/2)))*((- 3/52 - 1i/26)/d^2)^(1/2) - 2*atanh((4*d*((- 3/52 + 1i/26)/d^2)^(1/2) + 6*d *tan(c + d*x)*((- 3/52 + 1i/26)/d^2)^(1/2))/(tan(c + d*x)^(1/2)*(- 3*tan(c + d*x) - 2)^(1/2)))*((- 3/52 + 1i/26)/d^2)^(1/2)
\[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {-3 \tan \left (d x +c \right )-2}}{3 \tan \left (d x +c \right )^{2}+2 \tan \left (d x +c \right )}d x \right ) \] Input:
int(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x)
Output:
- int((sqrt(tan(c + d*x))*sqrt( - 3*tan(c + d*x) - 2))/(3*tan(c + d*x)**2 + 2*tan(c + d*x)),x)