Integrand size = 21, antiderivative size = 274 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {9 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {b^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d} \]
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Time = 0.30 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 21, 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} \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {9 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {6 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a b^3 \csc (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}-\frac {b^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {b^4 \sec (c+d x)}{d} \]
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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^4 \csc ^5(c+d x)+4 a^3 b \csc ^4(c+d x) \sec (c+d x)+6 a^2 b^2 \csc ^3(c+d x) \sec ^2(c+d x)+4 a b^3 \csc ^2(c+d x) \sec ^3(c+d x)+b^4 \csc (c+d x) \sec ^4(c+d x)\right ) \, dx \\ & = a^4 \int \csc ^5(c+d x) \, dx+\left (4 a^3 b\right ) \int \csc ^4(c+d x) \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \csc ^2(c+d x) \sec ^3(c+d x) \, dx+b^4 \int \csc (c+d x) \sec ^4(c+d x) \, dx \\ & = -\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {x^4}{-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^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {1}{8} \left (3 a^4\right ) \int \csc (c+d x) \, dx-\frac {\left (4 a^3 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^2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 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 (9 a^2 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {9 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {b^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1491\) vs. \(2(274)=548\).
Time = 8.07 (sec) , antiderivative size = 1491, normalized size of antiderivative = 5.44 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {b^2 \left (36 a^2+7 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 {\left (-7 a^3 b \cos \left (\frac {1}{2} (c+d x)\right )-6 a b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x) \csc \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 \left (a^4+8 a^2 b^2\right ) \cos ^4(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^3 b \cos ^4(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))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^4 \cos ^4(c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{64 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-3 a^4-72 a^2 b^2-8 b^4\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}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+3 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 (3 a^4+72 a^2 b^2+8 b^4\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}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+3 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 {3 \left (a^4+8 a^2 b^2\right ) \cos ^4(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {a^4 \cos ^4(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{64 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) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-7 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-6 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{3 d (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 )-7 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 )+7 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 {a^3 b \cos ^4(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))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4} \]
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Time = 17.35 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(241\) |
default | \(\frac {b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(241\) |
risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (216 a^{2} b^{2}+9 a^{4}+24 b^{4}+216 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-216 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+144 i a \,b^{3}+192 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+544 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-96 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}-144 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+128 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}-128 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+48 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-544 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-180 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+288 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+96 i a^{3} b +288 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-216 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-192 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+24 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(778\) |
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Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (264) = 528\).
Time = 0.40 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.00 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {6 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 16 \, {\left (18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\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 (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32 \, {\left (3 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 3 \, a b^{3} \cos \left (d x + c\right ) - 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.41 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.11 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 72 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 48 \, a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, b^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 32 \, a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
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Time = 1.09 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.75 \[ \int \csc ^5(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 )^{4} - 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 384 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 24 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {256 \, {\left (3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, 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} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a^{2} b^{2} - 2 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - \frac {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 4.90 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.13 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \]
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