Integrand size = 27, antiderivative size = 135 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {15 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \cot ^2(c+d x)}{2 d}-\frac {15 b \csc (c+d x)}{8 d}+\frac {3 a \log (\tan (c+d x))}{d}+\frac {5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {3 a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^4(c+d x)}{4 d} \]
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Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2913, 2700, 272, 45, 2701, 294, 327, 213} \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \tan ^4(c+d x)}{4 d}+\frac {3 a \tan ^2(c+d x)}{2 d}-\frac {a \cot ^2(c+d x)}{2 d}+\frac {3 a \log (\tan (c+d x))}{d}+\frac {15 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {15 b \csc (c+d x)}{8 d}+\frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {5 b \csc (c+d x) \sec ^2(c+d x)}{8 d} \]
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Rule 45
Rule 213
Rule 272
Rule 294
Rule 327
Rule 2700
Rule 2701
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \csc ^3(c+d x) \sec ^5(c+d x) \, dx+b \int \csc ^2(c+d x) \sec ^5(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^3} \, dx,x,\tan (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^3} \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {a \text {Subst}\left (\int \frac {(1+x)^3}{x^2} \, dx,x,\tan ^2(c+d x)\right )}{2 d}-\frac {(5 b) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{4 d} \\ & = \frac {5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {a \text {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d}-\frac {(15 b) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{8 d} \\ & = -\frac {a \cot ^2(c+d x)}{2 d}-\frac {15 b \csc (c+d x)}{8 d}+\frac {3 a \log (\tan (c+d x))}{d}+\frac {5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {3 a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^4(c+d x)}{4 d}-\frac {(15 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{8 d} \\ & = \frac {15 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \cot ^2(c+d x)}{2 d}-\frac {15 b \csc (c+d x)}{8 d}+\frac {3 a \log (\tan (c+d x))}{d}+\frac {5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac {b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac {3 a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^4(c+d x)}{4 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.74 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {b \csc (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},3,\frac {1}{2},\sin ^2(c+d x)\right )}{d}-\frac {3 a \log (\cos (c+d x))}{d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {a \sec ^2(c+d x)}{d}+\frac {a \sec ^4(c+d x)}{4 d} \]
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Time = 0.97 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(129\) |
default | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(129\) |
parallelrisch | \(\frac {-12 \left (a +\frac {5 b}{8}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-12 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -\frac {5 b}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+12 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {27 \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )+\frac {2 \cos \left (4 d x +4 c \right )}{9}-\frac {\cos \left (6 d x +6 c \right )}{9}+\frac {2}{27}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {80 b \left (\cos \left (2 d x +2 c \right )+\frac {3 \cos \left (4 d x +4 c \right )}{8}+\frac {9}{40}\right )}{27}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(247\) |
risch | \(-\frac {i \left (24 i a \,{\mathrm e}^{10 i \left (d x +c \right )}+15 b \,{\mathrm e}^{11 i \left (d x +c \right )}+48 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+25 b \,{\mathrm e}^{9 i \left (d x +c \right )}-16 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-22 b \,{\mathrm e}^{7 i \left (d x +c \right )}+48 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+22 b \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-25 b \,{\mathrm e}^{3 i \left (d x +c \right )}-15 b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(266\) |
norman | \(\frac {-\frac {a}{8 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {13 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {5 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {5 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {13 b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {21 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {21 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {27 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {27 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 \left (8 a -5 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}-\frac {3 \left (8 a +5 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}\) | \(317\) |
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Time = 0.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.56 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {24 \, a \cos \left (d x + c\right )^{4} - 12 \, a \cos \left (d x + c\right )^{2} + 48 \, {\left (a \cos \left (d x + c\right )^{6} - a \cos \left (d x + c\right )^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 3 \, {\left ({\left (8 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{6} - {\left (8 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (8 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{6} - {\left (8 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, b \cos \left (d x + c\right )^{4} - 5 \, b \cos \left (d x + c\right )^{2} - 2 \, b\right )} \sin \left (d x + c\right ) - 4 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{4}\right )}} \]
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Timed out. \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.04 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {3 \, {\left (8 \, a - 5 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (8 \, a + 5 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - 48 \, a \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, b \sin \left (d x + c\right )^{5} + 12 \, a \sin \left (d x + c\right )^{4} - 25 \, b \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 8 \, b \sin \left (d x + c\right ) + 4 \, a\right )}}{\sin \left (d x + c\right )^{6} - 2 \, \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{2}}}{16 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {3 \, {\left (8 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (8 \, a + 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 48 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {2 \, {\left (15 \, b \sin \left (d x + c\right )^{5} + 12 \, a \sin \left (d x + c\right )^{4} - 25 \, b \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 8 \, b \sin \left (d x + c\right ) + 4 \, a\right )}}{{\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{16 \, d} \]
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Time = 11.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,a}{2}-\frac {15\,b}{16}\right )}{d}-\frac {\frac {15\,b\,{\sin \left (c+d\,x\right )}^5}{8}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {25\,b\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {9\,a\,{\sin \left (c+d\,x\right )}^2}{4}+b\,\sin \left (c+d\,x\right )+\frac {a}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,a}{2}+\frac {15\,b}{16}\right )}{d} \]
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