Integrand size = 27, antiderivative size = 165 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {\left (8 a^3+9 a^2 b-b^3\right ) \log (1-\sin (c+d x))}{16 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {\left (8 a^3-9 a^2 b+b^3\right ) \log (1+\sin (c+d x))}{16 d}+\frac {\sec ^2(c+d x) \left (4 a^3+b \left (9 a^2-b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d} \]
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Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2916, 12, 1819, 837, 815} \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3 \log (\sin (c+d x))}{d}+\frac {\sec ^4(c+d x) \left (b \left (3 a^2+b^2\right ) \sin (c+d x)+a \left (a^2+3 b^2\right )\right )}{4 d}-\frac {\left (8 a^3+9 a^2 b-b^3\right ) \log (1-\sin (c+d x))}{16 d}-\frac {\left (8 a^3-9 a^2 b+b^3\right ) \log (\sin (c+d x)+1)}{16 d}+\frac {\sec ^2(c+d x) \left (4 a^3+b \left (9 a^2-b^2\right ) \sin (c+d x)\right )}{8 d} \]
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Rule 12
Rule 815
Rule 837
Rule 1819
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {b (a+x)^3}{x \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^6 \text {Subst}\left (\int \frac {(a+x)^3}{x \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}-\frac {b^4 \text {Subst}\left (\int \frac {-4 a^3-\left (9 a^2-b^2\right ) x}{x \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = \frac {\sec ^2(c+d x) \left (4 a^3+b \left (9 a^2-b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}-\frac {\text {Subst}\left (\int \frac {-8 a^3 b^2-b^2 \left (9 a^2-b^2\right ) x}{x \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {\sec ^2(c+d x) \left (4 a^3+b \left (9 a^2-b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}-\frac {\text {Subst}\left (\int \left (\frac {-8 a^3-9 a^2 b+b^3}{2 (b-x)}-\frac {8 a^3}{x}+\frac {8 a^3-9 a^2 b+b^3}{2 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = -\frac {\left (8 a^3+9 a^2 b-b^3\right ) \log (1-\sin (c+d x))}{16 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {\left (8 a^3-9 a^2 b+b^3\right ) \log (1+\sin (c+d x))}{16 d}+\frac {\sec ^2(c+d x) \left (4 a^3+b \left (9 a^2-b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.95 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-\left (\left (8 a^3+9 a^2 b-b^3\right ) \log (1-\sin (c+d x))\right )+16 a^3 \log (\sin (c+d x))-\left (8 a^3-9 a^2 b+b^3\right ) \log (1+\sin (c+d x))+\frac {(a+b)^3}{(-1+\sin (c+d x))^2}-\frac {(5 a-b) (a+b)^2}{-1+\sin (c+d x)}+\frac {(a-b)^3}{(1+\sin (c+d x))^2}+\frac {(a-b)^2 (5 a+b)}{1+\sin (c+d x)}}{16 d} \]
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Time = 1.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {3 a \,b^{2}}{4 \cos \left (d x +c \right )^{4}}+b^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(167\) |
default | \(\frac {a^{3} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {3 a \,b^{2}}{4 \cos \left (d x +c \right )^{4}}+b^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(167\) |
parallelrisch | \(\frac {-32 \left (a^{2}+\frac {1}{8} a b -\frac {1}{8} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-32 \left (a^{2}-\frac {1}{8} a b -\frac {1}{8} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-8 a^{3}-24 a \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\left (-6 a^{3}-6 a \,b^{2}\right ) \cos \left (4 d x +4 c \right )+\left (18 a^{2} b -2 b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (66 a^{2} b +14 b^{3}\right ) \sin \left (d x +c \right )+14 a^{3}+30 a \,b^{2}}{8 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(270\) |
risch | \(\frac {i \left (-8 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-9 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-32 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-48 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-33 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-7 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-8 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+33 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+9 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{3}}{d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{3}}{8 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{3}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(345\) |
norman | \(\frac {\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (4 a^{3}+24 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (4 a^{3}+24 a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (8 a^{3}+18 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (8 a^{3}+18 a \,b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \left (21 a^{2}+11 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \left (15 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {b \left (15 a^{2}+b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {b \left (27 a^{2}+5 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {b \left (27 a^{2}+5 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {b \left (81 a^{2}+31 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {b \left (81 a^{2}+31 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 a^{3}-9 a^{2} b +b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}-\frac {\left (8 a^{3}+9 a^{2} b -b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}\) | \(468\) |
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Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {16 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (8 \, a^{3} - 9 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{3} + 9 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} + 12 \, a b^{2} + 2 \, {\left (6 \, a^{2} b + 2 \, b^{3} + {\left (9 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.97 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {16 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - {\left (8 \, a^{3} - 9 \, a^{2} b + b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{3} + 9 \, a^{2} b - b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (4 \, a^{3} \sin \left (d x + c\right )^{2} + {\left (9 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{3} - 6 \, a^{3} - 6 \, a b^{2} - {\left (15 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {16 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - {\left (8 \, a^{3} - 9 \, a^{2} b + b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (8 \, a^{3} + 9 \, a^{2} b - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (6 \, a^{3} \sin \left (d x + c\right )^{4} - 9 \, a^{2} b \sin \left (d x + c\right )^{3} + b^{3} \sin \left (d x + c\right )^{3} - 16 \, a^{3} \sin \left (d x + c\right )^{2} + 15 \, a^{2} b \sin \left (d x + c\right ) + b^{3} \sin \left (d x + c\right ) + 12 \, a^{3} + 6 \, a b^{2}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 11.83 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02 \[ \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {a^3}{2}-\frac {9\,a^2\,b}{16}+\frac {b^3}{16}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {a^3}{2}+\frac {9\,a^2\,b}{16}-\frac {b^3}{16}\right )}{d}+\frac {\frac {3\,a\,b^2}{4}-{\sin \left (c+d\,x\right )}^3\,\left (\frac {9\,a^2\,b}{8}-\frac {b^3}{8}\right )+\frac {3\,a^3}{4}+\sin \left (c+d\,x\right )\,\left (\frac {15\,a^2\,b}{8}+\frac {b^3}{8}\right )-\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )} \]
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