Integrand size = 26, antiderivative size = 354 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {4 i \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}} \]
1/16*x*2^(2/3)/a^(4/3)-1/16*I*ln(cos(d*x+c))*2^(2/3)/a^(4/3)/d+2/3*I*ln(ta n(d*x+c))/a^(4/3)/d-2*I*ln(a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))/a^(4/3)/d-3/1 6*I*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^(2/3)/a^(4/3)/d-4/3*I*a rctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))/a^(4/3)/d* 3^(1/2)-1/8*I*arctan(1/3*(a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3 )*3^(1/2))*3^(1/2)*2^(2/3)/a^(4/3)/d-11/8*I/d/(a+I*a*tan(d*x+c))^(4/3)-cot (d*x+c)/d/(a+I*a*tan(d*x+c))^(4/3)-19/4*I/a/d/(a+I*a*tan(d*x+c))^(1/3)
Time = 3.39 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {\frac {64 i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac {6 i 2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {32 i \log (\tan (c+d x))}{a^{4/3}}-\frac {3 i 2^{2/3} \log (i+\tan (c+d x))}{a^{4/3}}+\frac {96 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3}}+\frac {9 i 2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3}}+\frac {66 i}{(a+i a \tan (c+d x))^{4/3}}+\frac {48 \cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}}+\frac {228 i}{a \sqrt [3]{a+i a \tan (c+d x)}}}{48 d} \]
-1/48*(((64*I)*Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/( Sqrt[3]*a^(1/3))])/a^(4/3) + ((6*I)*2^(2/3)*Sqrt[3]*ArcTan[(1 + (2^(2/3)*( a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(4/3) - ((32*I)*Log[Tan[ c + d*x]])/a^(4/3) - ((3*I)*2^(2/3)*Log[I + Tan[c + d*x]])/a^(4/3) + ((96* I)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/a^(4/3) + ((9*I)*2^(2/3)*L og[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/a^(4/3) + (66*I)/(a + I*a*Tan[c + d*x])^(4/3) + (48*Cot[c + d*x])/(a + I*a*Tan[c + d*x])^(4/3) + (228*I)/(a*(a + I*a*Tan[c + d*x])^(1/3)))/d
Time = 1.44 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.92, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4044, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4083, 3042, 3962, 67, 16, 1082, 217, 4082, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^2 (a+i a \tan (c+d x))^{4/3}}dx\) |
| \(\Big \downarrow \) 4044 |
| \(\displaystyle \frac {\int -\frac {\cot (c+d x) (7 \tan (c+d x) a+4 i a)}{3 (i \tan (c+d x) a+a)^{4/3}}dx}{a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) (7 \tan (c+d x) a+4 i a)}{(i \tan (c+d x) a+a)^{4/3}}dx}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {7 \tan (c+d x) a+4 i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {3 \int \frac {4 \cot (c+d x) \left (11 \tan (c+d x) a^2+8 i a^2\right )}{3 \sqrt [3]{i \tan (c+d x) a+a}}dx}{8 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left (11 \tan (c+d x) a^2+8 i a^2\right )}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {11 \tan (c+d x) a^2+8 i a^2}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {1}{3} \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (19 \tan (c+d x) a^3+16 i a^3\right )dx}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (19 \tan (c+d x) a^3+16 i a^3\right )dx}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {(i \tan (c+d x) a+a)^{2/3} \left (19 \tan (c+d x) a^3+16 i a^3\right )}{\tan (c+d x)}dx}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 4083 |
| \(\displaystyle -\frac {\frac {\frac {3 a^3 \int (i \tan (c+d x) a+a)^{2/3}dx+16 i a^2 \int \cot (c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}dx}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 a^3 \int (i \tan (c+d x) a+a)^{2/3}dx+16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle -\frac {\frac {\frac {16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^4 \int \frac {1}{(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {\frac {\frac {16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^4 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {\frac {\frac {16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^4 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\frac {16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^4 \left (\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {16 i a^2 \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle -\frac {\frac {\frac {\frac {16 i a^4 \int \frac {\cot (c+d x)}{\sqrt [3]{i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {\frac {\frac {\frac {16 i a^4 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {\frac {\frac {\frac {16 i a^4 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\frac {\frac {16 i a^4 \left (-\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {57 i a^2}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {\frac {16 i a^4 \left (\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^4 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}}{2 a^2}+\frac {33 i a}{8 d (a+i a \tan (c+d x))^{4/3}}}{3 a}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}\) |
-(Cot[c + d*x]/(d*(a + I*a*Tan[c + d*x])^(4/3))) - ((((33*I)/8)*a)/(d*(a + I*a*Tan[c + d*x])^(4/3)) + ((((-3*I)*a^4*(((-I)*Sqrt[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(1/3)*a^(1/3)) - (3*Log[2^(1/3)*a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(1/3)*a^(1/3)) + Log[a - I*a*Tan[c + d*x]]/(2*2^(1/3)*a^(1/3) )))/d + ((16*I)*a^4*((Sqrt[3]*ArcTan[(1 + (2*(a + I*a*Tan[c + d*x])^(1/3)) /a^(1/3))/Sqrt[3]])/a^(1/3) - Log[Tan[c + d*x]]/(2*a^(1/3)) + (3*Log[a^(1/ 3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(1/3))))/d)/(2*a^2) + (((57*I)/2) *a^2)/(d*(a + I*a*Tan[c + d*x])^(1/3)))/(2*a^2))/(3*a)
3.4.6.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
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[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d S ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b , c, d, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(c^2 + d^2)*(n + 1)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d , e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( A*b + a*B)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A *d)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Time = 0.82 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {3 i a^{3} \left (-\frac {5}{4 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}-\frac {1}{8 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{4}}-\frac {\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}}{4 a^{4}}\right )}{d}\) | \(315\) |
| default | \(\frac {3 i a^{3} \left (-\frac {5}{4 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}-\frac {1}{8 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{4}}-\frac {\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}}{4 a^{4}}\right )}{d}\) | \(315\) |
3*I/d*a^3*(-5/4/a^4/(a+I*a*tan(d*x+c))^(1/3)-1/8/a^3/(a+I*a*tan(d*x+c))^(4 /3)+1/a^4*(1/3*I*(a+I*a*tan(d*x+c))^(2/3)/a/tan(d*x+c)-4/9/a^(1/3)*ln((a+I *a*tan(d*x+c))^(1/3)-a^(1/3))+2/9/a^(1/3)*ln((a+I*a*tan(d*x+c))^(2/3)+a^(1 /3)*(a+I*a*tan(d*x+c))^(1/3)+a^(2/3))-4/9*3^(1/2)/a^(1/3)*arctan(1/3*3^(1/ 2)*(2/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))-1/4*(1/6*2^(2/3)/a^(1/3)*ln((a +I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/12*2^(2/3)/a^(1/3)*ln((a+I*a*tan (d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+2^(2/3)*a^(2/3))+1 /6*3^(1/2)*2^(2/3)/a^(1/3)*arctan(1/3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan( d*x+c))^(1/3)+1)))/a^4)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (252) = 504\).
Time = 0.27 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.24 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\text {Too large to display} \]
1/32*(2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(-95*I*e^(6*I*d*x + 6*I* c) - 35*I*e^(4*I*d*x + 4*I*c) + 63*I*e^(2*I*d*x + 2*I*c) + 3*I)*e^(4/3*I*d *x + 4/3*I*c) + 32*(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c)) *(64/27*I/(a^4*d^3))^(1/3)*log(9/16*a^3*d^2*(64/27*I/(a^4*d^3))^(2/3) + 2^ (1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 32*(a ^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d^3))^ (1/3)*log(32*a^3*d^2*(1/128*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((-I*sqrt(3)*a^2*d + a^2* d)*e^(6*I*d*x + 6*I*c) + (I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(6 4/27*I/(a^4*d^3))^(1/3)*log(-9/32*(I*sqrt(3)*a^3*d^2 + a^3*d^2)*(64/27*I/( a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (-I*sqr t(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(64/27*I/(a^4*d^3))^(1/3)*log(-9/ 32*(-I*sqrt(3)*a^3*d^2 + a^3*d^2)*(64/27*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/( e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((-I*sqrt(3) *a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d^3))^(1/3)*log(-16*(I*sqrt(3)*a^3*d^2 + a^3*d^2) *(1/128*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e ^(2/3*I*d*x + 2/3*I*c)) - 16*((I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c ) + (-I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d^3))...
\[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {i \, a {\left (\frac {6 \, {\left (38 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 27 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a - 3 \, a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{3}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{3}} + \frac {6 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {3 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {6 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}} + \frac {64 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {32 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {64 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}}\right )}}{48 \, d} \]
-1/48*I*a*(6*(38*(I*a*tan(d*x + c) + a)^2 - 27*(I*a*tan(d*x + c) + a)*a - 3*a^2)/((I*a*tan(d*x + c) + a)^(7/3)*a^2 - (I*a*tan(d*x + c) + a)^(4/3)*a^ 3) + 6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I* a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(7/3) - 3*2^(2/3)*log(2^(2/3)*a^(2/3 ) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^ (2/3))/a^(7/3) + 6*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^( 1/3))/a^(7/3) + 64*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1 /3) + a^(1/3))/a^(1/3))/a^(7/3) - 32*log((I*a*tan(d*x + c) + a)^(2/3) + (I *a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 64*log((I*a*tan(d* x + c) + a)^(1/3) - a^(1/3))/a^(7/3))/d
\[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\cot \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
Time = 7.44 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.52 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\text {Too large to display} \]
log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i - ((46656*a^7*d^6*(64i/(27*a^4*d ^3))^(2/3) + 55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3))*(64i/(27*a^4*d^3 ))^(1/3) - a^3*d^3*37107i)*(64i/(27*a^4*d^3))^(2/3))*(64i/(27*a^4*d^3))^(1 /3) - (((a + a*tan(c + d*x)*1i)*27i)/(8*d) + (a*3i)/(8*d) - ((a + a*tan(c + d*x)*1i)^2*19i)/(4*a*d))/(a*(a + a*tan(c + d*x)*1i)^(4/3) - (a + a*tan(c + d*x)*1i)^(7/3)) + log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i - ((46656*a ^7*d^6*(1i/(128*a^4*d^3))^(2/3) + 55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1 /3))*(1i/(128*a^4*d^3))^(1/3) - a^3*d^3*37107i)*(1i/(128*a^4*d^3))^(2/3))* (1i/(128*a^4*d^3))^(1/3) + (log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i + (( 3^(1/2)*1i - 1)^2*(a^3*d^3*37107i - ((3^(1/2)*1i - 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*(3^(1/2)*1i - 1)^2*(64i/(27*a^4*d ^3))^(2/3))*(64i/(27*a^4*d^3))^(1/3))/2)*(64i/(27*a^4*d^3))^(2/3))/4)*(3^( 1/2)*1i - 1)*(64i/(27*a^4*d^3))^(1/3))/2 - (log(d*(a + a*tan(c + d*x)*1i)^ (1/3)*1584i + ((3^(1/2)*1i + 1)^2*(a^3*d^3*37107i + ((3^(1/2)*1i + 1)*(557 82*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*(3^(1/2)*1i + 1)^ 2*(64i/(27*a^4*d^3))^(2/3))*(64i/(27*a^4*d^3))^(1/3))/2)*(64i/(27*a^4*d^3) )^(2/3))/4)*(3^(1/2)*1i + 1)*(64i/(27*a^4*d^3))^(1/3))/2 + (log(d*(a + a*t an(c + d*x)*1i)^(1/3)*1584i + ((3^(1/2)*1i - 1)^2*(a^3*d^3*37107i - ((3^(1 /2)*1i - 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*( 3^(1/2)*1i - 1)^2*(1i/(128*a^4*d^3))^(2/3))*(1i/(128*a^4*d^3))^(1/3))/2...