Integrand size = 25, antiderivative size = 233 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {(A-C) \arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {(A-C) \arctan \left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}} \]
-1/2*(A-C)*arctan(1-2^(1/2)*(b*tan(d*x+c))^(1/2)/b^(1/2))/d*2^(1/2)/b^(1/2 )+1/2*(A-C)*arctan(1+2^(1/2)*(b*tan(d*x+c))^(1/2)/b^(1/2))/d*2^(1/2)/b^(1/ 2)-1/4*(A-C)*ln(b^(1/2)-2^(1/2)*(b*tan(d*x+c))^(1/2)+b^(1/2)*tan(d*x+c))/d *2^(1/2)/b^(1/2)+1/4*(A-C)*ln(b^(1/2)+2^(1/2)*(b*tan(d*x+c))^(1/2)+b^(1/2) *tan(d*x+c))/d*2^(1/2)/b^(1/2)-2/3*b*C/d/(b*tan(d*x+c))^(3/2)
Time = 0.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.64 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {-8 C \cot (c+d x)-3 \sqrt {2} (A-C) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right ) \sqrt {\tan (c+d x)}}{12 d \sqrt {b \tan (c+d x)}} \]
(-8*C*Cot[c + d*x] - 3*Sqrt[2]*(A - C)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[T an[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12*d*Sqrt[b*Tan[c + d*x]])
Time = 0.66 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.92, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4156, 3042, 4112, 27, 2030, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {-b \cot \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle b^2 \int \frac {A \tan ^2(c+d x)+C}{(b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \int \frac {A \tan (c+d x)^2+C}{(b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4112 |
| \(\displaystyle b^2 \left (\frac {\int \frac {b (A-C) \tan (c+d x)}{(b \tan (c+d x))^{3/2}}dx}{b^2}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^2 \left (\frac {(A-C) \int \frac {\tan (c+d x)}{(b \tan (c+d x))^{3/2}}dx}{b}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle b^2 \left (\frac {(A-C) \int \frac {1}{\sqrt {b \tan (c+d x)}}dx}{b^2}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \left (\frac {(A-C) \int \frac {1}{\sqrt {b \tan (c+d x)}}dx}{b^2}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle b^2 \left (\frac {(A-C) \int \frac {1}{\sqrt {b \tan (c+d x)} \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \int \frac {1}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\int \frac {b-b^2 \tan ^2(c+d x)}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{2 b}+\frac {\int \frac {b^2 \tan ^2(c+d x)+b}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\frac {1}{2} \int \frac {1}{b^2 \tan ^2(c+d x)-\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}+\frac {1}{2} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 b}+\frac {\int \frac {b-b^2 \tan ^2(c+d x)}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\frac {\int \frac {1}{-b^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}-\frac {\int \frac {1}{-b^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}}{2 b}+\frac {\int \frac {b-b^2 \tan ^2(c+d x)}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\int \frac {b-b^2 \tan ^2(c+d x)}{b^4 \tan ^4(c+d x)+b^2}d\sqrt {b \tan (c+d x)}}{2 b}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {b}-2 \sqrt {b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {2} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {b}+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {2} \sqrt {b}}}{2 b}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {b}-2 \sqrt {b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {b}+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {2} \sqrt {b}}}{2 b}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {b}-2 \sqrt {b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {\sqrt {b}+\sqrt {2} \sqrt {b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {2} b^{3/2} \tan (c+d x)+b}d\sqrt {b \tan (c+d x)}}{2 \sqrt {b}}}{2 b}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle b^2 \left (\frac {2 (A-C) \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {b} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {b}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {b} \tan (c+d x)\right )}{\sqrt {2} \sqrt {b}}}{2 b}+\frac {\frac {\log \left (\sqrt {2} b^{3/2} \tan (c+d x)+b^2 \tan ^2(c+d x)+b\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\log \left (-\sqrt {2} b^{3/2} \tan (c+d x)+b^2 \tan ^2(c+d x)+b\right )}{2 \sqrt {2} \sqrt {b}}}{2 b}\right )}{b d}-\frac {2 C}{3 b d (b \tan (c+d x))^{3/2}}\right )\) |
b^2*((2*(A - C)*((-(ArcTan[1 - Sqrt[2]*Sqrt[b]*Tan[c + d*x]]/(Sqrt[2]*Sqrt [b])) + ArcTan[1 + Sqrt[2]*Sqrt[b]*Tan[c + d*x]]/(Sqrt[2]*Sqrt[b]))/(2*b) + (-1/2*Log[b - Sqrt[2]*b^(3/2)*Tan[c + d*x] + b^2*Tan[c + d*x]^2]/(Sqrt[2 ]*Sqrt[b]) + Log[b + Sqrt[2]*b^(3/2)*Tan[c + d*x] + b^2*Tan[c + d*x]^2]/(2 *Sqrt[2]*Sqrt[b]))/(2*b)))/(b*d) - (2*C)/(3*b*d*(b*Tan[c + d*x])^(3/2)))
3.1.1.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[ e + f*x])^(m + 1)*Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[ a^2 + b^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Time = 2.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(\frac {2 b \left (-\frac {C}{3 \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (A -C \right ) \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{2}}\right )}{d}\) | \(160\) |
| default | \(\frac {2 b \left (-\frac {C}{3 \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (A -C \right ) \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{2}}\right )}{d}\) | \(160\) |
| parts | \(\frac {A \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d b}+\frac {C \sqrt {2}\, \left (2 \left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \csc \left (d x +c \right )-3 \ln \left (-\frac {-\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )-2+2 \cos \left (d x +c \right )+\sin \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-6 \arctan \left (\frac {\sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+3 \ln \left (\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )+2-2 \cos \left (d x +c \right )-\sin \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-6 \arctan \left (\frac {\sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )+1-\cos \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )\right )}{12 d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \left (1-\cos \left (d x +c \right )\right ) \sqrt {-\frac {b \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(763\) |
2/d*b*(-1/3*C/(b*tan(d*x+c))^(3/2)+1/8*(A-C)*(b^2)^(1/4)/b^2*2^(1/2)*(ln(( b*tan(d*x+c)+(b^2)^(1/4)*(b*tan(d*x+c))^(1/2)*2^(1/2)+(b^2)^(1/2))/(b*tan( d*x+c)-(b^2)^(1/4)*(b*tan(d*x+c))^(1/2)*2^(1/2)+(b^2)^(1/2)))+2*arctan(2^( 1/2)/(b^2)^(1/4)*(b*tan(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(b^2)^(1/4)*(b* tan(d*x+c))^(1/2)+1)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.00 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {3 \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} + 3 i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} - 3 i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} - 3 \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} + 4 \, \sqrt {b \tan \left (d x + c\right )} C}{6 \, b d \tan \left (d x + c\right )^{2}} \]
-1/6*(3*b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4) *log(b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4) - sqrt(b*tan(d*x + c))*(A - C))*tan(d*x + c)^2 + 3*I*b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*log(I*b*d*(-(A^4 - 4*A^3*C + 6 *A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4) - sqrt(b*tan(d*x + c))*(A - C)) *tan(d*x + c)^2 - 3*I*b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b ^2*d^4))^(1/4)*log(-I*b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b ^2*d^4))^(1/4) - sqrt(b*tan(d*x + c))*(A - C))*tan(d*x + c)^2 - 3*b*d*(-(A ^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*log(-b*d*(-(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4) - sqrt(b*tan(d*x + c))*(A - C))*tan(d*x + c)^2 + 4*sqrt(b*tan(d*x + c))*C)/(b*d*tan(d*x + c )^2)
\[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\int \frac {A + C \cot ^{2}{\left (c + d x \right )}}{\sqrt {b \tan {\left (c + d x \right )}}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.77 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {3 \, {\left (2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right ) - \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )\right )} {\left (A - C\right )} - \frac {8 \, C b^{2}}{\left (b \tan \left (d x + c\right )\right )^{\frac {3}{2}}}}{12 \, b d} \]
1/12*(3*(2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(b) + 2*sqrt(b* tan(d*x + c)))/sqrt(b)) + 2*sqrt(2)*sqrt(b)*arctan(-1/2*sqrt(2)*(sqrt(2)*s qrt(b) - 2*sqrt(b*tan(d*x + c)))/sqrt(b)) + sqrt(2)*sqrt(b)*log(b*tan(d*x + c) + sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(b) + b) - sqrt(2)*sqrt(b)*log(b*t an(d*x + c) - sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(b) + b))*(A - C) - 8*C*b^2 /(b*tan(d*x + c))^(3/2))/(b*d)
Time = 0.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.06 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {2 \, C}{3 \, \sqrt {b \tan \left (d x + c\right )} d \tan \left (d x + c\right )} \]
1/2*sqrt(2)*(A*sqrt(abs(b)) - C*sqrt(abs(b)))*arctan(1/2*sqrt(2)*(sqrt(2)* sqrt(abs(b)) + 2*sqrt(b*tan(d*x + c)))/sqrt(abs(b)))/(b*d) + 1/2*sqrt(2)*( A*sqrt(abs(b)) - C*sqrt(abs(b)))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*tan(d*x + c)))/sqrt(abs(b)))/(b*d) + 1/4*sqrt(2)*(A*sqrt(abs(b )) - C*sqrt(abs(b)))*log(b*tan(d*x + c) + sqrt(2)*sqrt(b*tan(d*x + c))*sqr t(abs(b)) + abs(b))/(b*d) - 1/4*sqrt(2)*(A*sqrt(abs(b)) - C*sqrt(abs(b)))* log(b*tan(d*x + c) - sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b)) + abs(b))/( b*d) - 2/3*C/(sqrt(b*tan(d*x + c))*d*tan(d*x + c))
Time = 13.65 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.55 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {2\,C\,b}{3\,d\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}}{\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}}\right )\,\left (A-C\right )\,1{}\mathrm {i}}{\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}}{\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}}\right )\,\left (A-C\right )}{\sqrt {b}\,d} \]
((-1)^(1/4)*atan((((-1)^(1/4)*(A - C)*((b*tan(c + d*x))^(1/2)*(16*A^2*b^2* d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) - ((-1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 32*C*b^3*d^4))/(2*b^(1/2)*d))*1i)/(2*b^(1/2)*d) + ((-1)^(1/4)*(A - C)*( (b*tan(c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) + ((-1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 32*C*b^3*d^4))/(2*b^(1/2)*d))*1i)/(2 *b^(1/2)*d))/(((-1)^(1/4)*(A - C)*((b*tan(c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) - ((-1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 3 2*C*b^3*d^4))/(2*b^(1/2)*d)))/(2*b^(1/2)*d) - ((-1)^(1/4)*(A - C)*((b*tan( c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) + ((-1) ^(1/4)*(A - C)*(32*A*b^3*d^4 - 32*C*b^3*d^4))/(2*b^(1/2)*d)))/(2*b^(1/2)*d )))*(A - C)*1i)/(b^(1/2)*d) - (2*C*b)/(3*d*(b*tan(c + d*x))^(3/2)) + ((-1) ^(1/4)*atan((((-1)^(1/4)*(A - C)*((b*tan(c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) - ((-1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 32 *C*b^3*d^4)*1i)/(2*b^(1/2)*d)))/(2*b^(1/2)*d) + ((-1)^(1/4)*(A - C)*((b*ta n(c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) + ((- 1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 32*C*b^3*d^4)*1i)/(2*b^(1/2)*d)))/(2*b^(1 /2)*d))/(((-1)^(1/4)*(A - C)*((b*tan(c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16* C^2*b^2*d^3 - 32*A*C*b^2*d^3) - ((-1)^(1/4)*(A - C)*(32*A*b^3*d^4 - 32*C*b ^3*d^4)*1i)/(2*b^(1/2)*d))*1i)/(2*b^(1/2)*d) - ((-1)^(1/4)*(A - C)*((b*tan (c + d*x))^(1/2)*(16*A^2*b^2*d^3 + 16*C^2*b^2*d^3 - 32*A*C*b^2*d^3) + (...