Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}} \, dx=\frac {i \text {arctanh}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
I*arctanh((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*arctanh((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 = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}} \, dx=\frac {i \arctan \left (\frac {\sqrt {-3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {-3+2 i} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
(I*ArcTan[(Sqrt[-3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/( Sqrt[-3 + 2*I]*d) - (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)
Time = 0.26 (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, 221}
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 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {3 \tan (c+d x)+2}}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{(i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {3 \tan (c+d x)+2}}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {\int \frac {1}{\frac {(2-3 i) \tan (c+d x)}{3 \tan (c+d x)+2}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)+2}}-\int \frac {1}{i-\frac {(2+3 i) \tan (c+d x)}{3 \tan (c+d x)+2}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)+2}}}{d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {i \text {arctanh}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)+2}}\right )}{\sqrt {3-2 i}}-\frac {i \text {arctanh}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)+2}}\right )}{\sqrt {3+2 i}}}{d}\) |
((I*ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/ Sqrt[3 - 2*I] - (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*T an[c + d*x]]])/Sqrt[3 + 2*I])/d
3.7.63.3.1 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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. \(478\) vs. \(2(77)=154\).
Time = 4.27 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.04
method | result | size |
derivativedivides | \(-\frac {\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 {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-3 \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-12 \,\operatorname {arctanh}\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 \,\operatorname {arctanh}\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 )}\, \sqrt {2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(479\) |
default | \(-\frac {\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 {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-3 \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-12 \,\operatorname {arctanh}\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 \,\operatorname {arctanh}\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 )}\, \sqrt {2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(479\) |
-1/2/d*(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))*((2*13^(1/2)-6)^(1/2)*13^(1/2)*(2*13^(1/2)+6)^(1/2) *arctan(1/416*(2*13^(1/2)-6)^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(2+3*tan(d *x+c))*(39+11*13^(1/2))/(13^(1/2)-3+2*tan(d*x+c))^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))/tan( d*x+c)/(2+3*tan(d*x+c)))-3*(2*13^(1/2)-6)^(1/2)*(2*13^(1/2)+6)^(1/2)*arcta n(1/416*(2*13^(1/2)-6)^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(2+3*tan(d*x+c)) *(39+11*13^(1/2))/(13^(1/2)-3+2*tan(d*x+c))^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))/tan(d*x+c) /(2+3*tan(d*x+c)))-12*arctanh(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*arctanh (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)))/tan(d*x+c)^(1/2)/(2+3*tan(d*x+c))^(1/2)/(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 = 0.34 (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)*sq rt(3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqr t(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*ta n(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {3 \tan {\left (c + d x \right )} + 2}}\, dx \]
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {3 \, \tan \left (d x + c\right ) + 2}} \,d x } \]
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:Unable to find common minimal polyn omial Error: Bad Argument ValueDone
Time = 6.65 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}} \, dx=\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}\,\left (-6+4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}+2}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}\,\left (-6-4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}+2}\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]
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)*(3*tan(c + d*x) + 2)^(1 /2)*(6 - 4i))/(3*tan(c + d*x) - 2^(1/2)*(3*tan(c + d*x) + 2)^(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)*(3*tan(c + d*x) + 2)^(1/2)*(6 + 4i))/(3*tan(c + d*x) - 2^(1/2)*( 3*tan(c + d*x) + 2)^(1/2) + 2))*((- 3/52 + 1i/26)/d^2)^(1/2)*2i