Integrand size = 31, antiderivative size = 146 \[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d} \]
-(A-I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))*(a-I*b)^(1/2)/d-(A+ I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))*(a+I*b)^(1/2)/d+2*A*(a+ b*tan(d*x+c))^(1/2)/d+2/3*B*(a+b*tan(d*x+c))^(3/2)/b/d
Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.96 \[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {-3 \sqrt {a-i b} b (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-3 \sqrt {a+i b} b (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 \sqrt {a+b \tan (c+d x)} (3 A b+a B+b B \tan (c+d x))}{3 b d} \]
(-3*Sqrt[a - I*b]*b*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I* b]] - 3*Sqrt[a + I*b]*b*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b + a*B + b*B*Tan[c + d*x]))/(3* b*d)
Time = 0.73 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 4075, 3042, 4011, 3042, 4022, 3042, 4020, 25, 73, 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.
| \(\displaystyle \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \int (A \tan (c+d x)-B) \sqrt {a+b \tan (c+d x)}dx+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (A \tan (c+d x)-B) \sqrt {a+b \tan (c+d x)}dx+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {-A b-a B+(a A-b B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {-A b-a B+(a A-b B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {1}{2} (-b+i a) (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} (b+i a) (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} (-b+i a) (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} (b+i a) (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {i (b+i a) (A-i B) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i (-b+i a) (A+i B) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i (b+i a) (A-i B) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {i (-b+i a) (A+i B) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-b+i a) (A+i B) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {(b+i a) (A-i B) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(b+i a) (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {(-b+i a) (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d}\) |
-(((I*a + b)*(A - I*B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]* d)) + ((I*a - b)*(A + I*B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I *b]*d) + (2*A*Sqrt[a + b*Tan[c + d*x]])/d + (2*B*(a + b*Tan[c + d*x])^(3/2 ))/(3*b*d)
3.4.19.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B *d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(828\) vs. \(2(122)=244\).
Time = 0.11 (sec) , antiderivative size = 829, normalized size of antiderivative = 5.68
| method | result | size |
| parts | \(\frac {A \left (2 \sqrt {a +b \tan \left (d x +c \right )}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d}+B \left (\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d b}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{4 d b}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )\) | \(829\) |
| derivativedivides | \(\frac {2 B \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}+\frac {2 A \sqrt {a +b \tan \left (d x +c \right )}}{d}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\) | \(989\) |
| default | \(\frac {2 B \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}+\frac {2 A \sqrt {a +b \tan \left (d x +c \right )}}{d}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\) | \(989\) |
A/d*(2*(a+b*tan(d*x+c))^(1/2)-1/4*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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))+(a-(a^2+b^2)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d *x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)) +1/4*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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))+((a^2+b^2)^(1/2)-a)/(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)))+B*(2/3/d/b*(a+b*tan(d*x+c)) ^(3/2)-1/4/d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*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))+1/4/d/b*(2*(a ^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*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))-1/d*b/(2*(a^2+b^2)^ (1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^ (1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))+1/4/d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/2 )*a*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))-1/4/d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*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))+1/d*b/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 1228 vs. \(2 (116) = 232\).
Time = 0.29 (sec) , antiderivative size = 1228, normalized size of antiderivative = 8.41 \[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
1/6*(3*b*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a *b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3* B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*d^3*sqrt(-(4*A ^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + ( 2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)* a)/d^2)) - 3*b*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A* B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2 *(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - (B*d^3*sqrt (-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^ 4) + (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^ 2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)) - 3*b*d*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3* B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*l og(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*d^ 3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b ^2)/d^4) - (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b - d^2*sqrt(-(4* A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)) + 3*b*d*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4 *(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*...
\[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx \]
\[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 14.08 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.92 \[ \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (B^2\,b^4-B^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {-\frac {\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2}{d^4}}}{16\,\left (B^3\,a^2\,b^3+B^3\,b^5\right )}\right )\,\sqrt {-\frac {\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2}{d^4}}+\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (B^2\,b^4-B^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}-\frac {16\,a\,b^2\,\left (\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2}{d^4}}}{16\,\left (B^3\,a^2\,b^3+B^3\,b^5\right )}\right )\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2}{d^4}}-2\,\mathrm {atanh}\left (\frac {32\,A^2\,b^4\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}+\frac {A^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d^3}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d^3}}+\frac {32\,a\,b^2\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}+\frac {A^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-A^4\,b^2\,d^4}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d}}\right )\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}+A^2\,a\,d^2}{4\,d^4}}+2\,\mathrm {atanh}\left (\frac {32\,A^2\,b^4\,\sqrt {\frac {A^2\,a}{4\,d^2}-\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d^3}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d^3}}-\frac {32\,a\,b^2\,\sqrt {\frac {A^2\,a}{4\,d^2}-\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-A^4\,b^2\,d^4}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d}}\right )\,\sqrt {-\frac {\sqrt {-A^4\,b^2\,d^4}-A^2\,a\,d^2}{4\,d^4}}+\frac {2\,A\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {2\,B\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,b\,d} \]
atanh((d^3*((16*(B^2*b^4 - B^2*a^2*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 + (16*a*b^2*((-B^4*b^2*d^4)^(1/2) + B^2*a*d^2)*(a + b*tan(c + d*x))^(1/2))/d ^4)*(-((-B^4*b^2*d^4)^(1/2) + B^2*a*d^2)/d^4)^(1/2))/(16*(B^3*b^5 + B^3*a^ 2*b^3)))*(-((-B^4*b^2*d^4)^(1/2) + B^2*a*d^2)/d^4)^(1/2) + atanh((d^3*((16 *(B^2*b^4 - B^2*a^2*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 - (16*a*b^2*((-B^ 4*b^2*d^4)^(1/2) - B^2*a*d^2)*(a + b*tan(c + d*x))^(1/2))/d^4)*(((-B^4*b^2 *d^4)^(1/2) - B^2*a*d^2)/d^4)^(1/2))/(16*(B^3*b^5 + B^3*a^2*b^3)))*(((-B^4 *b^2*d^4)^(1/2) - B^2*a*d^2)/d^4)^(1/2) - 2*atanh((32*A^2*b^4*((-A^4*b^2*d ^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((1 6*A*b^4*(-A^4*b^2*d^4)^(1/2))/d^3 + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d^ 3) + (32*a*b^2*((-A^4*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-A^4*b^2*d^4)^(1/2))/((16*A*b^4*(-A^4*b^2*d^4)^(1/ 2))/d + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d))*(((-A^4*b^2*d^4)^(1/2) + A ^2*a*d^2)/(4*d^4))^(1/2) + 2*atanh((32*A^2*b^4*((A^2*a)/(4*d^2) - (-A^4*b^ 2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A*b^4*(-A^4*b ^2*d^4)^(1/2))/d^3 + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d^3) - (32*a*b^2* ((A^2*a)/(4*d^2) - (-A^4*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*tan(c + d*x) )^(1/2)*(-A^4*b^2*d^4)^(1/2))/((16*A*b^4*(-A^4*b^2*d^4)^(1/2))/d + (16*A*a ^2*b^2*(-A^4*b^2*d^4)^(1/2))/d))*(-((-A^4*b^2*d^4)^(1/2) - A^2*a*d^2)/(4*d ^4))^(1/2) + (2*A*(a + b*tan(c + d*x))^(1/2))/d + (2*B*(a + b*tan(c + d...