Integrand size = 12, antiderivative size = 107 \[ \int (a+b \csc (c+d x))^4 \, dx=a^4 x-\frac {2 a b \left (2 a^2+b^2\right ) \text {arctanh}(\cos (c+d x))}{d}-\frac {b^2 \left (17 a^2+2 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d} \]
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Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3867, 4133, 3855, 3852, 8} \[ \int (a+b \csc (c+d x))^4 \, dx=a^4 x-\frac {2 a b \left (2 a^2+b^2\right ) \text {arctanh}(\cos (c+d x))}{d}-\frac {b^2 \left (17 a^2+2 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3867
Rule 4133
Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d}+\frac {1}{3} \int (a+b \csc (c+d x)) \left (3 a^3+b \left (9 a^2+2 b^2\right ) \csc (c+d x)+8 a b^2 \csc ^2(c+d x)\right ) \, dx \\ & = -\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2+b^2\right ) \csc (c+d x)+2 b^2 \left (17 a^2+2 b^2\right ) \csc ^2(c+d x)\right ) \, dx \\ & = a^4 x-\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d}+\left (2 a b \left (2 a^2+b^2\right )\right ) \int \csc (c+d x) \, dx+\frac {1}{3} \left (b^2 \left (17 a^2+2 b^2\right )\right ) \int \csc ^2(c+d x) \, dx \\ & = a^4 x-\frac {2 a b \left (2 a^2+b^2\right ) \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d}-\frac {\left (b^2 \left (17 a^2+2 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d} \\ & = a^4 x-\frac {2 a b \left (2 a^2+b^2\right ) \text {arctanh}(\cos (c+d x))}{d}-\frac {b^2 \left (17 a^2+2 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac {b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(568\) vs. \(2(107)=214\).
Time = 12.92 (sec) , antiderivative size = 568, normalized size of antiderivative = 5.31 \[ \int (a+b \csc (c+d x))^4 \, dx=\frac {a^4 (c+d x) (a+b \csc (c+d x))^4 \sin ^4(c+d x)}{d (b+a \sin (c+d x))^4}+\frac {\left (-9 a^2 b^2 \cos \left (\frac {1}{2} (c+d x)\right )-b^4 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right ) (a+b \csc (c+d x))^4 \sin ^4(c+d x)}{3 d (b+a \sin (c+d x))^4}-\frac {a b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \csc (c+d x))^4 \sin ^4(c+d x)}{2 d (b+a \sin (c+d x))^4}-\frac {b^4 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \csc (c+d x))^4 \sin ^4(c+d x)}{24 d (b+a \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+a b^3\right ) (a+b \csc (c+d x))^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^4(c+d x)}{d (b+a \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+a b^3\right ) (a+b \csc (c+d x))^4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^4(c+d x)}{d (b+a \sin (c+d x))^4}+\frac {a b^3 (a+b \csc (c+d x))^4 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^4(c+d x)}{2 d (b+a \sin (c+d x))^4}+\frac {(a+b \csc (c+d x))^4 \sec \left (\frac {1}{2} (c+d x)\right ) \left (9 a^2 b^2 \sin \left (\frac {1}{2} (c+d x)\right )+b^4 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^4(c+d x)}{3 d (b+a \sin (c+d x))^4}+\frac {b^4 (a+b \csc (c+d x))^4 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^4(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d (b+a \sin (c+d x))^4} \]
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Time = 1.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {a^{4} \left (d x +c \right )+4 a^{3} b \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-6 a^{2} b^{2} \cot \left (d x +c \right )+4 a \,b^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+b^{4} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(112\) |
default | \(\frac {a^{4} \left (d x +c \right )+4 a^{3} b \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-6 a^{2} b^{2} \cot \left (d x +c \right )+4 a \,b^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+b^{4} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(112\) |
parts | \(a^{4} x +\frac {b^{4} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}-\frac {2 a \,b^{3} \cot \left (d x +c \right ) \csc \left (d x +c \right )}{d}+\frac {2 a \,b^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}-\frac {6 a^{2} b^{2} \cot \left (d x +c \right )}{d}-\frac {4 a^{3} b \ln \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{d}\) | \(118\) |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}-\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}-12 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+24 a^{4} x d +72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}-72 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}-9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}+96 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}}{24 d}\) | \(173\) |
norman | \(\frac {a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {b^{4}}{24 d}+\frac {b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{24 d}-\frac {a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 d}-\frac {3 b^{2} \left (8 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 d}+\frac {3 b^{2} \left (8 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2 a b \left (2 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(178\) |
risch | \(a^{4} x +\frac {4 b^{2} \left (-9 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+18 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 i a^{2}-i b^{2}-3 a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(196\) |
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (101) = 202\).
Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.03 \[ \int (a+b \csc (c+d x))^4 \, dx=-\frac {2 \, {\left (9 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} b + a b^{3} - {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (2 \, a^{3} b + a b^{3} - {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right ) - 3 \, {\left (a^{4} d x \cos \left (d x + c\right )^{2} - a^{4} d x + 2 \, a b^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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\[ \int (a+b \csc (c+d x))^4 \, dx=\int \left (a + b \csc {\left (c + d x \right )}\right )^{4}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17 \[ \int (a+b \csc (c+d x))^4 \, dx=a^{4} x + \frac {a b^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{d} - \frac {4 \, a^{3} b \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{d} - \frac {6 \, a^{2} b^{2}}{d \tan \left (d x + c\right )} - \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} b^{4}}{3 \, d \tan \left (d x + c\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (101) = 202\).
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.92 \[ \int (a+b \csc (c+d x))^4 \, dx=\frac {b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, {\left (d x + c\right )} a^{4} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, {\left (2 \, a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {176 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 88 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 18.00 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.93 \[ \int (a+b \csc (c+d x))^4 \, dx=\frac {b^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {b^4\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {3\,b^4\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {3\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {2\,a^4\,\mathrm {atan}\left (\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}{-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )}{d}+\frac {2\,a\,b^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^3\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,a^2\,b^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}-\frac {a\,b^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {3\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}+\frac {a\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d} \]
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