Integrand size = 23, antiderivative size = 272 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
8/5*a*b^2/d/cot(d*x+c)^(3/2)+2/5*b^2*(b+a*cot(d*x+c))/d/cot(d*x+c)^(5/2)-1 /2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/2 *(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a -b)*(a^2+4*a*b+b^2)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/ 4*(a-b)*(a^2+4*a*b+b^2)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2 )+2*b*(3*a^2-b^2)/d/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.38 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 \left (\left (-9 a^2 b+3 b^3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )+a \left (a (9 b+5 a \cot (c+d x))-5 \left (a^2-3 b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{15 d \cot ^{\frac {5}{2}}(c+d x)} \]
(2*((-9*a^2*b + 3*b^3)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + a*(a*(9*b + 5*a*Cot[c + d*x]) - 5*(a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometr ic2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*Cot[c + d*x]^(5/2))
Time = 0.99 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.93, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4111, 27, 3042, 4012, 3042, 4017, 25, 1482, 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.
\(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}}dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \int \frac {(a \cot (c+d x)+b)^3}{\cot ^{\frac {7}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4048 |
\(\displaystyle \frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {12 a b^2+5 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (5 a^2-3 b^2\right ) \cot ^2(c+d x)}{2 \cot ^{\frac {5}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {12 a b^2+5 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (5 a^2-3 b^2\right ) \cot ^2(c+d x)}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \int \frac {12 a b^2-5 \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b+a \left (5 a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4111 |
\(\displaystyle \frac {1}{5} \left (\int \frac {5 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (5 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (5 \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {1}{5} \left (5 \left (\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (5 \left (\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \int -\frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}-\frac {2 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{5} \left (5 \left (\frac {2 \left (-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {8 a b^2}{d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
(2*b^2*(b + a*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2)) + ((8*a*b^2)/(d*Cot[ c + d*x]^(3/2)) + 5*((2*b*(3*a^2 - b^2))/(d*Sqrt[Cot[c + d*x]]) + (2*(-1/2 *((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sq rt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) - ((a - b)*(a^2 + 4*a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sq rt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2 ))/d))/5
3.9.18.3.1 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[((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[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (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*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + 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[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + 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 = 1.36 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(246\) |
default | \(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(246\) |
-1/d*(-2/5*b^3/cot(d*x+c)^(5/2)-2*b*(3*a^2-b^2)/cot(d*x+c)^(1/2)-2*a*b^2/c ot(d*x+c)^(3/2)+1/4*(a^3-3*a*b^2)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d* x+c)^(1/2))/(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*co t(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/4*(-3*a^2*b+b^3)* 2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+2^(1/2)* cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2 )*cot(d*x+c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1399 vs. \(2 (234) = 468\).
Time = 0.36 (sec) , antiderivative size = 1399, normalized size of antiderivative = 5.14 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\text {Too large to display} \]
-1/10*(5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^1 0*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) )/d^2)*log(((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b ^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^ 5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^ 8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27 *a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) - 5*d*sqrt((6*a^5*b - 2 0*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^ 6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((a^3 - 3*a*b^2) *d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^ 7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^ 10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4 ))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^ 12)*sqrt(tan(d*x + c))) - 5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*s qrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^ 2*b^10 + b^12)/d^4))/d^2)*log(((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b ^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b...
\[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{3}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {8 \, {\left (b^{3} + \frac {5 \, a b^{2}}{\tan \left (d x + c\right )} + \frac {5 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{20 \, d} \]
1/20*(8*(b^3 + 5*a*b^2/tan(d*x + c) + 5*(3*a^2*b - b^3)/tan(d*x + c)^2)*ta n(d*x + c)^(5/2) - 10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*s qrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) - 10*sqrt(2)*(a^3 - 3*a^2*b - 3*a *b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 5*sqrt (2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan (d*x + c) + 1) + 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sq rt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/d
\[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]