Integrand size = 21, antiderivative size = 229 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {9 a b^2 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {3 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {3 b^3 \csc (c+d x)}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a b^2 \sec (c+d x)}{2 d}-\frac {3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d} \]
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Time = 0.25 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3598, 3853, 3855, 2701, 308, 213, 2702, 294, 327} \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {9 a b^2 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {9 a b^2 \sec (c+d x)}{2 d}-\frac {3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 b^3 \csc (c+d x)}{2 d}+\frac {b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d} \]
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Rule 213
Rule 294
Rule 308
Rule 327
Rule 2701
Rule 2702
Rule 3598
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \csc ^5(c+d x)+3 a^2 b \csc ^4(c+d x) \sec (c+d x)+3 a b^2 \csc ^3(c+d x) \sec ^2(c+d x)+b^3 \csc ^2(c+d x) \sec ^3(c+d x)\right ) \, dx \\ & = a^3 \int \csc ^5(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^4(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc ^2(c+d x) \sec ^3(c+d x) \, dx \\ & = -\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}+\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (9 a b^2\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = -\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {3 b^3 \csc (c+d x)}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a b^2 \sec (c+d x)}{2 d}-\frac {3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (9 a b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = -\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {9 a b^2 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {3 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {3 b^3 \csc (c+d x)}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a b^2 \sec (c+d x)}{2 d}-\frac {3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1229\) vs. \(2(229)=458\).
Time = 7.64 (sec) , antiderivative size = 1229, normalized size of antiderivative = 5.37 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {\left (-7 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )-2 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^3(c+d x) \csc \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {3 \left (a^3+4 a b^2\right ) \cos ^3(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {a^2 b \cos ^3(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {a^3 \cos ^3(c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {3 \left (a^3+12 a b^2\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {3 \left (2 a^2 b+b^3\right ) \cos ^3(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))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 \left (a^3+12 a b^2\right ) \cos ^3(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 \left (2 a^2 b+b^3\right ) \cos ^3(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))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 \left (a^3+4 a b^2\right ) \cos ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {a^3 \cos ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}+\frac {3 a b^2 \cos ^3(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{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))^3}-\frac {b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}-\frac {3 a b^2 \cos ^3(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{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))^3}+\frac {\cos ^3(c+d x) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-7 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {a^2 b \cos ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3} \]
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Time = 10.94 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
default | \(\frac {b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-5 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+24 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+12 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+3 i a^{3}+36 i a \,b^{2}-56 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-28 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+36 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-80 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-60 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+80 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-24 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-60 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+56 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+28 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-5 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-24 a^{2} b -12 b^{3}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(580\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (211) = 422\).
Time = 0.36 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.86 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {6 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 48 \, a b^{2} \cos \left (d x + c\right ) - 10 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 12 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, {\left (3 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} + b^{3} - 4 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \csc ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.09 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \]
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Time = 0.76 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.63 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 32 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, {\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, {\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 24 \, {\left (a^{3} + 12 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {64 \, {\left (b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \]
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Time = 5.46 (sec) , antiderivative size = 698, normalized size of antiderivative = 3.05 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{8}+\frac {3\,a\,b^2}{8}\right )}{d}-\frac {\mathrm {atan}\left (\frac {\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{4}+9\,a\,b^2\right )+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )-6\,a^2\,b-3\,b^3\right )\,1{}\mathrm {i}-\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{4}+9\,a\,b^2\right )+6\,a^2\,b+3\,b^3\right )\,1{}\mathrm {i}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (36\,a^4\,b^2+36\,a^2\,b^4+9\,b^6\right )+27\,a\,b^5+\frac {9\,a^5\,b}{2}-\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{4}+9\,a\,b^2\right )+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )-6\,a^2\,b-3\,b^3\right )-\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^2\,b+\frac {3\,b^3}{2}\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{4}+9\,a\,b^2\right )+6\,a^2\,b+3\,b^3\right )+\frac {225\,a^3\,b^3}{4}}\right )\,\left (a^2\,b\,6{}\mathrm {i}+b^3\,3{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^3}{2}+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (2\,a^3+102\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {15\,a^3}{4}+108\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (26\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (30\,a^2\,b-8\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (58\,a^2\,b+32\,b^3\right )+\frac {a^3}{4}+2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,a^2\,b}{8}+\frac {b^3}{2}\right )}{d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+12\,b^2\right )}{8\,d}-\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d} \]
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