Integrand size = 27, antiderivative size = 162 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {59 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {11 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))} \]
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Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}-\frac {59 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {11 a \log (\sin (c+d x)+1)}{16 d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^5 \text {Subst}\left (\int \frac {a^4}{(a-x)^3 x^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^9 \text {Subst}\left (\int \frac {1}{(a-x)^3 x^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^9 \text {Subst}\left (\int \left (\frac {1}{4 a^6 (a-x)^3}+\frac {5}{4 a^7 (a-x)^2}+\frac {59}{16 a^8 (a-x)}+\frac {1}{a^5 x^4}+\frac {1}{a^6 x^3}+\frac {3}{a^7 x^2}+\frac {3}{a^8 x}+\frac {1}{8 a^7 (a+x)^2}+\frac {11}{16 a^8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {3 a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {59 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {11 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.64 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},3,-\frac {1}{2},\sin ^2(c+d x)\right )}{3 d}-\frac {3 a \log (\cos (c+d x))}{d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {a \sec ^2(c+d x)}{d}+\frac {a \sec ^4(c+d x)}{4 d} \]
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Time = 0.68 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(147\) |
default | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(147\) |
risch | \(-\frac {i a \left (-138 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{11 i \left (d x +c \right )}+136 i {\mathrm e}^{8 i \left (d x +c \right )}-101 \,{\mathrm e}^{9 i \left (d x +c \right )}+260 i {\mathrm e}^{6 i \left (d x +c \right )}-158 \,{\mathrm e}^{7 i \left (d x +c \right )}+136 i {\mathrm e}^{4 i \left (d x +c \right )}+158 \,{\mathrm e}^{5 i \left (d x +c \right )}-138 i {\mathrm e}^{2 i \left (d x +c \right )}+101 \,{\mathrm e}^{3 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {59 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(233\) |
parallelrisch | \(-\frac {\left (59 \left (2-\sin \left (3 d x +3 c \right )-\sin \left (d x +c \right )+2 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+11 \left (\sin \left (3 d x +3 c \right )+\sin \left (d x +c \right )-2 \cos \left (2 d x +2 c \right )-2\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \left (\sin \left (3 d x +3 c \right )+\sin \left (d x +c \right )-2 \cos \left (2 d x +2 c \right )-2\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+26 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\left (\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{12}+\frac {\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{12}+\frac {\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{12}+\frac {\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )}{12}\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-488 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2195 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {606 \cos \left (2 d x +2 c \right )}{2195}+\frac {101 \cos \left (3 d x +3 c \right )}{2195}-\frac {610}{439}\right )}{3}\right ) a}{8 d \left (2-\sin \left (3 d x +3 c \right )-\sin \left (d x +c \right )+2 \cos \left (2 d x +2 c \right )\right )}\) | \(339\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (150) = 300\).
Time = 0.27 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.12 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {138 \, a \cos \left (d x + c\right )^{4} - 172 \, a \cos \left (d x + c\right )^{2} - 144 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 33 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 177 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, a \cos \left (d x + c\right )^{4} - 104 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 18 \, a}{48 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {33 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 177 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (105 \, a \sin \left (d x + c\right )^{5} - 69 \, a \sin \left (d x + c\right )^{4} - 106 \, a \sin \left (d x + c\right )^{3} + 52 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 8 \, a\right )}}{\sin \left (d x + c\right )^{6} - \sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{3}}}{48 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.92 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {66 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 354 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 288 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {6 \, {\left (11 \, a \sin \left (d x + c\right ) + 13 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {3 \, {\left (177 \, a \sin \left (d x + c\right )^{2} - 394 \, a \sin \left (d x + c\right ) + 221 \, a\right )}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {16 \, {\left (33 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{96 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {35\,a\,{\sin \left (c+d\,x\right )}^5}{8}-\frac {23\,a\,{\sin \left (c+d\,x\right )}^4}{8}-\frac {53\,a\,{\sin \left (c+d\,x\right )}^3}{12}+\frac {13\,a\,{\sin \left (c+d\,x\right )}^2}{6}+\frac {a\,\sin \left (c+d\,x\right )}{6}+\frac {a}{3}}{d\,\left ({\sin \left (c+d\,x\right )}^6-{\sin \left (c+d\,x\right )}^5-{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^3\right )}-\frac {59\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{16\,d}+\frac {11\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{16\,d} \]
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