Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}-\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
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)
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=-\frac {i \left (\sqrt {3+2 i} \arctan \left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \text {arctanh}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \]
((-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]*S qrt[-2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4058, 613, 104, 218}
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.
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)} (\tan (c+d x)+i)}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\int \frac {1}{i-\frac {(2-3 i) \tan (c+d x)}{-3 \tan (c+d x)-2}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}-\int \frac {1}{\frac {(2+3 i) \tan (c+d x)}{-3 \tan (c+d x)-2}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i}}-\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i}}}{d}\) |
((I*ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]])/ Sqrt[3 - 2*I] - (I*ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*T an[c + d*x]]])/Sqrt[3 + 2*I])/d
3.7.66.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[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] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(e_.)*(x_)]/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Sym bol] :> Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] + x)), x ], x] - Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] - x)), x ], x] /; FreeQ[{a, b, c, d, e}, x]
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. \(437\) vs. \(2(77)=154\).
Time = 4.01 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.61
| method | result | size |
| derivativedivides | \(\frac {\sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {2 \sqrt {13}-6}\, \operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-3 \sqrt {2 \sqrt {13}+6}\, \sqrt {2 \sqrt {13}-6}\, \operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(438\) |
| default | \(\frac {\sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {2 \sqrt {13}-6}\, \operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-3 \sqrt {2 \sqrt {13}+6}\, \sqrt {2 \sqrt {13}-6}\, \operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(438\) |
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*tan(d*x+c))*(13^(1/2)*(2*13^(1/2)+6)^(1 /2)*(2*13^(1/2)-6)^(1/2)*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))-3* (2*13^(1/2)+6)^(1/2)*(2*13^(1/2)-6)^(1/2)*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))+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*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^2)^(1/2)/(2 6*13^(1/2)+78)^(1/2)))/tan(d*x+c)^(1/2)/(2+3*tan(d*x+c))/(2*13^(1/2)+6)^(1 /2)/(11*13^(1/2)-39)
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 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=\text {Too large to display} \]
-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*ta n(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*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*t an(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)...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {- 3 \tan {\left (c + d x \right )} - 2}}\, dx \]
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {-3 \, \tan \left (d x + c\right ) - 2}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Time = 7.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {12\,d\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{3}\right )}{\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}\,\left (\frac {3\,{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2}{3\,\mathrm {tan}\left (c+d\,x\right )+2}+1\right )}\right )\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {12\,d\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}{\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}\,\left (\frac {3\,{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}{3\,\mathrm {tan}\left (c+d\,x\right )+2}+1\right )}\right )\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]
atan((12*d*((3/52 - 1i/26)/d^2)^(1/2)*((6^(1/2)*1i)/3 - tan(c + d*x)^(1/2) ))/((- 3*tan(c + d*x) - 2)^(1/2)*((3*((6^(1/2)*1i)/3 - tan(c + d*x)^(1/2)) ^2)/(3*tan(c + d*x) + 2) + 1)))*((3/52 - 1i/26)/d^2)^(1/2)*2i - atan((12*d *((3/52 + 1i/26)/d^2)^(1/2)*((2^(1/2)*3^(1/2)*1i)/3 - tan(c + d*x)^(1/2))) /((- 3*tan(c + d*x) - 2)^(1/2)*((3*((2^(1/2)*3^(1/2)*1i)/3 - tan(c + d*x)^ (1/2))^2)/(3*tan(c + d*x) + 2) + 1)))*((3/52 + 1i/26)/d^2)^(1/2)*2i