Integrand size = 23, antiderivative size = 424 \[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \sqrt {a+b \tan (c+d x)}}{b d} \]
-1/2*b*arctanh(((a+(a^2+b^2)^(1/2))^(1/2)-2^(1/2)*(a+b*tan(d*x+c))^(1/2))/ (a-(a^2+b^2)^(1/2))^(1/2))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a-(a^2+b^2)^(1/2))^( 1/2)+1/2*b*arctanh(((a+(a^2+b^2)^(1/2))^(1/2)+2^(1/2)*(a+b*tan(d*x+c))^(1/ 2))/(a-(a^2+b^2)^(1/2))^(1/2))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a-(a^2+b^2)^(1/2 ))^(1/2)+1/4*b*ln(a+(a^2+b^2)^(1/2)-2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a+b *tan(d*x+c))^(1/2)+b*tan(d*x+c))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a+(a^2+b^2)^(1 /2))^(1/2)-1/4*b*ln(a+(a^2+b^2)^(1/2)+2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a +b*tan(d*x+c))^(1/2)+b*tan(d*x+c))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a+(a^2+b^2)^ (1/2))^(1/2)+2*(a+b*tan(d*x+c))^(1/2)/b/d
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.25 \[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}+\frac {2 \sqrt {a+b \tan (c+d x)}}{b}}{d} \]
((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*Ar cTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]])/b)/d
Time = 0.76 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4026, 25, 3042, 3966, 484, 1407, 1142, 25, 27, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^2}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int -\frac {1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 \sqrt {a+b \tan (c+d x)}}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\int \frac {1}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\int \frac {1}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {b \int \frac {1}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 484 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \int \frac {1}{b^4 \tan ^4(c+d x)-2 a b^2 \tan ^2(c+d x)+a^2+b^2}d\sqrt {a+b \tan (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-\sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+\sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {1}{2} \int -\frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-b^2 \tan ^2(c+d x)}d\left (2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-b^2 \tan ^2(c+d x)}d\left (\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \sqrt {a+b \tan (c+d x)}}{b d}-\frac {2 b \left (\frac {-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}-\frac {1}{2} \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {1}{2} \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\) |
(-2*b*((-((Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(-(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]) + 2*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]) ])/Sqrt[a - Sqrt[a^2 + b^2]]) - Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2] *Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]) + (-((Sqrt[a + Sqrt[a^2 + b^2] ]*ArcTanh[(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]] + 2*Sqrt[a + b*Tan[c + d*x]]) /(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - Sqrt[a^2 + b^2]]) + Log[Sq rt[a^2 + b^2] + b^2*Tan[c + d*x]^2 + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqr t[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b ^2]])))/d + (2*Sqrt[a + b*Tan[c + d*x]])/(b*d)
3.6.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* d Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1576\) vs. \(2(343)=686\).
Time = 0.07 (sec) , antiderivative size = 1577, normalized size of antiderivative = 3.72
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1577\) |
| default | \(\text {Expression too large to display}\) | \(1577\) |
2*(a+b*tan(d*x+c))^(1/2)/b/d+1/4/d/b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*( 2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^ (1/2)+2*a)^(1/2)*a^2+1/4/d*b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b ^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2* a)^(1/2)-1/4/d/b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1 /2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/ 2)*a^3-1/4/d*b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2 )+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2) *a+3/d*b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2 )^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2) )*a^2+2/d*b^3/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^ 2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^ (1/2))-1/d/b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2 +b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^( 1/2))*a^2-1/d*b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*( a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a )^(1/2))+1/d/b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a ^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a) ^(1/2))*a^4-1/4/d/b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2* (a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (345) = 690\).
Time = 0.26 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {b d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b + {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + b^{2} d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - b d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b - {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + b^{2} d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - b d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b + {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - b^{2} d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) + b d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b - {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - b^{2} d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - 4 \, \sqrt {b \tan \left (d x + c\right ) + a}}{2 \, b d} \]
-1/2*(b*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b + ((a^3 + a*b^2)*d^ 3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b^2*d)*sqrt(-((a^2 + b^2)*d^2 *sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))) - b*d*s qrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b - ((a^3 + a*b^2)*d^3*sqrt(-b^2 /((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b^2*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/ ((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))) - b*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) *log(sqrt(b*tan(d*x + c) + a)*b + ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a ^2*b^2 + b^4)*d^4)) - b^2*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2 *b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))) + b*d*sqrt(((a^2 + b^2)*d^2*sqr t(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(sqrt(b*t an(d*x + c) + a)*b - ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4) *d^4)) - b^2*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d ^4)) - a)/((a^2 + b^2)*d^2))) - 4*sqrt(b*tan(d*x + c) + a))/(b*d)
\[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\tan ^{2}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{2}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]
Time = 6.15 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.72 \[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\ln \left (-16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}+\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}}{2}-\ln \left (16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}-\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{4\,\left (a\,d^2-b\,d^2\,1{}\mathrm {i}\right )}}+\frac {2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{b\,d}-\mathrm {atan}\left (-\frac {b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{-\frac {64\,a\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^4\,d^2\,64{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {128\,a\,b^3\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {a^2\,b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,2{}\mathrm {i} \]
(log(16*b^3*d*(-1/(d^2*(a - b*1i)))^(1/2) - 16*b^2*(a + b*tan(c + d*x))^(1 /2) + (16*a*b^2*(a + b*tan(c + d*x))^(1/2))/(a - b*1i))*(-1/(a*d^2 - b*d^2 *1i))^(1/2))/2 - log(16*b^2*(a + b*tan(c + d*x))^(1/2) + 16*b^3*d*(-1/(d^2 *(a - b*1i)))^(1/2) - (16*a*b^2*(a + b*tan(c + d*x))^(1/2))/(a - b*1i))*(- 1/(4*(a*d^2 - b*d^2*1i)))^(1/2) - atan((128*a*b^3*((b*1i)/(4*a^2*d^2 + 4*b ^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((b ^6*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (a^2*b^4*d^2*256i)/(4*a^2*d^3 + 4*b ^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a*b^5*d^2)/(4*a ^2*d^3 + 4*b^2*d^3)) - (b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^4*d^2*64i)/(4*a^2 *d^3 + 4*b^2*d^3) - (64*a*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) + (a^2*b^2*((b *1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan (c + d*x))^(1/2)*128i)/((b^6*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (a^2*b^4* d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d ^3) - (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*2i + (2*(a + b*tan(c + d*x))^(1/2))/(b*d)