Integrand size = 21, antiderivative size = 161 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {a^4 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {6 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d} \]
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Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3598, 3853, 3855, 2701, 327, 213, 2702, 2686, 30} \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {a^4 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}+\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d} \]
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Rule 30
Rule 213
Rule 327
Rule 2686
Rule 2701
Rule 2702
Rule 3598
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \csc ^3(c+d x)+4 a^3 b \csc ^2(c+d x) \sec (c+d x)+6 a^2 b^2 \csc (c+d x) \sec ^2(c+d x)+4 a b^3 \sec ^3(c+d x)+b^4 \sec ^3(c+d x) \tan (c+d x)\right ) \, dx \\ & = a^4 \int \csc ^3(c+d x) \, dx+\left (4 a^3 b\right ) \int \csc ^2(c+d x) \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^3(c+d x) \, dx+b^4 \int \sec ^3(c+d x) \tan (c+d x) \, dx \\ & = -\frac {a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d}+\frac {1}{2} a^4 \int \csc (c+d x) \, dx+\left (2 a b^3\right ) \int \sec (c+d x) \, dx-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d}-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {6 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1128\) vs. \(2(161)=322\).
Time = 8.04 (sec) , antiderivative size = 1128, normalized size of antiderivative = 7.01 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {b^2 \left (36 a^2+b^2\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 a^3 b \cos ^4(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^4 \cos ^4(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-a^4-12 a^2 b^2\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (a^4+12 a^2 b^2\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {a^4 \cos ^4(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (12 a b^3+b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {b^4 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {b^4 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-12 a b^3+b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (-36 a^2 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-b^4 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (36 a^2 b^2 \sin \left (\frac {1}{2} (c+d x)\right )+b^4 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 a^3 b \cos ^4(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4} \]
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Time = 6.39 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {b^{4}}{3 \cos \left (d x +c \right )^{3}}+4 a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(157\) |
default | \(\frac {\frac {b^{4}}{3 \cos \left (d x +c \right )^{3}}+4 a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(157\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (8 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+8 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+12 a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+36 i a^{2} b^{2}-16 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+48 a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-24 a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+18 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+12 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-72 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+36 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-48 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+24 a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{4}-24 a^{3} b -12 a \,b^{3}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(510\) |
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (155) = 310\).
Time = 0.33 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.15 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {6 \, {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, b^{4} - 4 \, {\left (18 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 12 \, {\left ({\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left ({\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 24 \, {\left (a b^{3} \cos \left (d x + c\right ) - {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.17 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a^{2} b^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 24 \, a^{3} b {\left (\frac {2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {4 \, b^{4}}{\cos \left (d x + c\right )^{3}}}{12 \, d} \]
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Time = 1.07 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.86 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, {\left (2 \, a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 48 \, {\left (2 \, a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 12 \, {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {3 \, {\left (6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{4}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {16 \, {\left (6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, a^{2} b^{2} - b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{24 \, d} \]
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Time = 5.10 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.16 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {\frac {a^4}{2}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^4}{2}+48\,a^2\,b^2+\frac {8\,b^4}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4}{2}+48\,a^2\,b^2+8\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^4}{2}+96\,a^2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (16\,a\,b^3-8\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (24\,a^3\,b+16\,a\,b^3\right )+8\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+24\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^4}{2}+6\,a^2\,b^2\right )}{d}-\frac {2\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}-\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\left (2\,a^2+b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+12\,a^2\,b^2\right )-4\,a\,b^3-8\,a^3\,b+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\right )\,2{}\mathrm {i}-a\,b\,\left (2\,a^2+b^2\right )\,\left (4\,a\,b^3-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+12\,a^2\,b^2\right )+8\,a^3\,b+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\right )\,2{}\mathrm {i}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6\,b^2+64\,a^4\,b^4+16\,a^2\,b^6\right )+8\,a^7\,b+48\,a^3\,b^5+100\,a^5\,b^3-2\,a\,b\,\left (2\,a^2+b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+12\,a^2\,b^2\right )-4\,a\,b^3-8\,a^3\,b+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\right )-2\,a\,b\,\left (2\,a^2+b^2\right )\,\left (4\,a\,b^3-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+12\,a^2\,b^2\right )+8\,a^3\,b+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\right )}\right )\,\left (2\,a^2+b^2\right )\,4{}\mathrm {i}}{d} \]
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