Integrand size = 29, antiderivative size = 221 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {3 (a+b) \left (8 a^2+7 a b+b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {3 a \left (a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {3 (a-b) \left (8 a^2-7 a b+b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {b^2 \sec ^4(c+d x) \left (a \left (3+\frac {a^2}{b^2}\right )+\left (1+\frac {3 a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {b^2 \sec ^2(c+d x) \left (4 a \left (3+\frac {2 a^2}{b^2}\right )+3 \left (1+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d} \]
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Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2916, 12, 1819, 1816} \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {3 (a+b) \left (8 a^2+7 a b+b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {3 a \left (a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {3 (a-b) \left (8 a^2-7 a b+b^2\right ) \log (\sin (c+d x)+1)}{16 d}+\frac {b^2 \sec ^4(c+d x) \left (b \left (\frac {3 a^2}{b^2}+1\right ) \sin (c+d x)+a \left (\frac {a^2}{b^2}+3\right )\right )}{4 d}+\frac {b^2 \sec ^2(c+d x) \left (3 b \left (\frac {7 a^2}{b^2}+1\right ) \sin (c+d x)+4 a \left (\frac {2 a^2}{b^2}+3\right )\right )}{8 d}-\frac {3 a^2 b \csc (c+d x)}{d} \]
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Rule 12
Rule 1816
Rule 1819
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {b^3 (a+x)^3}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^8 \text {Subst}\left (\int \frac {(a+x)^3}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^2 \sec ^4(c+d x) \left (a \left (3+\frac {a^2}{b^2}\right )+\left (1+\frac {3 a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}-\frac {b^6 \text {Subst}\left (\int \frac {-4 a^3-12 a^2 x-4 a \left (3+\frac {a^2}{b^2}\right ) x^2-3 \left (1+\frac {3 a^2}{b^2}\right ) x^3}{x^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = \frac {b^2 \sec ^4(c+d x) \left (a \left (3+\frac {a^2}{b^2}\right )+\left (1+\frac {3 a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {b^2 \sec ^2(c+d x) \left (4 a \left (3+\frac {2 a^2}{b^2}\right )+3 \left (1+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac {b^4 \text {Subst}\left (\int \frac {8 a^3+24 a^2 x+8 a \left (3+\frac {2 a^2}{b^2}\right ) x^2+3 \left (1+\frac {7 a^2}{b^2}\right ) x^3}{x^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {b^2 \sec ^4(c+d x) \left (a \left (3+\frac {a^2}{b^2}\right )+\left (1+\frac {3 a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {b^2 \sec ^2(c+d x) \left (4 a \left (3+\frac {2 a^2}{b^2}\right )+3 \left (1+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac {b^4 \text {Subst}\left (\int \left (\frac {3 (a+b) \left (8 a^2+7 a b+b^2\right )}{2 b^4 (b-x)}+\frac {8 a^3}{b^2 x^3}+\frac {24 a^2}{b^2 x^2}+\frac {24 a \left (a^2+b^2\right )}{b^4 x}+\frac {3 (a-b) \left (-8 a^2+7 a b-b^2\right )}{2 b^4 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = -\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {3 (a+b) \left (8 a^2+7 a b+b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {3 a \left (a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {3 (a-b) \left (8 a^2-7 a b+b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {b^2 \sec ^4(c+d x) \left (a \left (3+\frac {a^2}{b^2}\right )+\left (1+\frac {3 a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {b^2 \sec ^2(c+d x) \left (4 a \left (3+\frac {2 a^2}{b^2}\right )+3 \left (1+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d} \\ \end{align*}
Time = 2.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.86 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-48 a^2 b \csc (c+d x)-8 a^3 \csc ^2(c+d x)-3 (a+b) \left (8 a^2+7 a b+b^2\right ) \log (1-\sin (c+d x))+48 a \left (a^2+b^2\right ) \log (\sin (c+d x))-3 (a-b) \left (8 a^2-7 a b+b^2\right ) \log (1+\sin (c+d x))+\frac {(a+b)^3}{(-1+\sin (c+d x))^2}-\frac {3 (a+b)^2 (3 a+b)}{-1+\sin (c+d x)}+\frac {(a-b)^3}{(1+\sin (c+d x))^2}+\frac {3 (a-b)^2 (3 a-b)}{1+\sin (c+d x)}}{16 d} \]
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Time = 1.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{2} b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(217\) |
default | \(\frac {a^{3} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{2} b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(217\) |
parallelrisch | \(\frac {-12 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right ) \left (a^{2}+\frac {7}{8} a b +\frac {1}{8} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-12 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right ) \left (a^{2}-\frac {7}{8} a b +\frac {1}{8} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+12 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+b^{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {27 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (2 d x +2 c \right )+\frac {2 \cos \left (4 d x +4 c \right )}{9}-\frac {\cos \left (6 d x +6 c \right )}{9}+\frac {2}{27}\right ) a^{3} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {15 b \left (\cos \left (2 d x +2 c \right )+\frac {3 \cos \left (4 d x +4 c \right )}{8}+\frac {9}{40}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-3 b^{2} \left (a \cos \left (2 d x +2 c \right )+\frac {3 \cos \left (4 d x +4 c \right ) a}{4}-\frac {b \sin \left (3 d x +3 c \right )}{4}-\frac {11 b \sin \left (d x +c \right )}{12}-\frac {7 a}{4}\right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(338\) |
risch | \(-\frac {i \left (48 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-144 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+45 a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}+24 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+75 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+5 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+48 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-16 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-66 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-30 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+24 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+66 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+30 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+48 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+48 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-75 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-5 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{4 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 )^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{3}}{d}+\frac {45 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b}{8 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a \,b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{3}}{8 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {45 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b}{8 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a \,b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{3}}{8 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(577\) |
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Time = 0.29 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.52 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {24 \, {\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, a^{3} - 12 \, a b^{2} - 12 \, {\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left ({\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 3 \, {\left ({\left (8 \, a^{3} - 15 \, a^{2} b + 8 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (8 \, a^{3} - 15 \, a^{2} b + 8 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (8 \, a^{3} + 15 \, a^{2} b + 8 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (8 \, a^{3} + 15 \, a^{2} b + 8 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, {\left (15 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} - 6 \, a^{2} b - 2 \, b^{3} - {\left (15 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{4}\right )}} \]
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Timed out. \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.98 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 \, {\left (8 \, a^{3} - 15 \, a^{2} b + 8 \, a b^{2} - b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (8 \, a^{3} + 15 \, a^{2} b + 8 \, a b^{2} + b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - 48 \, {\left (a^{3} + a b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, {\left (15 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )^{5} + 12 \, {\left (a^{3} + a b^{2}\right )} \sin \left (d x + c\right )^{4} + 24 \, a^{2} b \sin \left (d x + c\right ) - 5 \, {\left (15 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )^{3} + 4 \, a^{3} - 18 \, {\left (a^{3} + a b^{2}\right )} \sin \left (d x + c\right )^{2}\right )}}{\sin \left (d x + c\right )^{6} - 2 \, \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{2}}}{16 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.09 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 \, {\left (8 \, a^{3} - 15 \, a^{2} b + 8 \, a b^{2} - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (8 \, a^{3} + 15 \, a^{2} b + 8 \, a b^{2} + b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 48 \, {\left (a^{3} + a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {2 \, {\left (45 \, a^{2} b \sin \left (d x + c\right )^{5} + 3 \, b^{3} \sin \left (d x + c\right )^{5} + 12 \, a^{3} \sin \left (d x + c\right )^{4} + 12 \, a b^{2} \sin \left (d x + c\right )^{4} - 75 \, a^{2} b \sin \left (d x + c\right )^{3} - 5 \, b^{3} \sin \left (d x + c\right )^{3} - 18 \, a^{3} \sin \left (d x + c\right )^{2} - 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 24 \, a^{2} b \sin \left (d x + c\right ) + 4 \, a^{3}\right )}}{{\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{16 \, d} \]
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Time = 11.64 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )\right )\,\left (a^2+b^2\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,a^3}{2}+\frac {3\,a\,b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {9\,a^3}{4}+\frac {9\,a\,b^2}{4}\right )+{\sin \left (c+d\,x\right )}^5\,\left (\frac {45\,a^2\,b}{8}+\frac {3\,b^3}{8}\right )-{\sin \left (c+d\,x\right )}^3\,\left (\frac {75\,a^2\,b}{8}+\frac {5\,b^3}{8}\right )+\frac {a^3}{2}+3\,a^2\,b\,\sin \left (c+d\,x\right )}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )}-\frac {3\,\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-b\right )\,\left (8\,a^2-7\,a\,b+b^2\right )}{16\,d}-\frac {3\,\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+b\right )\,\left (8\,a^2+7\,a\,b+b^2\right )}{16\,d} \]
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