Integrand size = 29, antiderivative size = 100 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d} \]
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Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2990, 2700, 14, 3280, 457, 79, 53, 65, 212} \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (3 a^2+2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x) \text {arctanh}\left (\sqrt {\cos ^2(c+d x)}\right )}{2 d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d} \]
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Rule 14
Rule 53
Rule 65
Rule 79
Rule 212
Rule 457
Rule 2700
Rule 2990
Rule 3280
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+\int \csc ^3(c+d x) \sec ^2(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {(2 a b) \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a^2+b^2 x^2}{x^3 \left (1-x^2\right )^{3/2}} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {(2 a b) \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a^2+b^2 x}{(1-x)^{3/2} x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {2 a b \cot (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(1-x)^{3/2} x} \, dx,x,\sin ^2(c+d x)\right )}{4 d} \\ & = -\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\sin ^2(c+d x)\right )}{4 d} \\ & = -\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}+\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos ^2(c+d x)}\right )}{2 d} \\ & = -\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {\left (3 a^2+2 b^2\right ) \text {arctanh}\left (\sqrt {\cos ^2(c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(100)=200\).
Time = 1.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.38 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\csc ^4(c+d x) \left (2 a^2+4 b^2-2 \left (3 a^2+2 b^2\right ) \cos (2 (c+d x))+3 a^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\left (3 a^2+2 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-3 a^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 a b \sin (c+d x)-8 a b \sin (3 (c+d x))\right )}{2 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.72 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(116\) |
default | \(\frac {a^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(116\) |
parallelrisch | \(\frac {\left (\left (12 a^{2}+8 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 a^{2}-8 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 a b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-48 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-18 a^{2}-16 b^{2}}{8 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 d}\) | \(149\) |
risch | \(\frac {3 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+3 a^{2} {\mathrm e}^{i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{i \left (d x +c \right )}-8 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+8 i a b}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(215\) |
norman | \(\frac {\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a^{2}}{8 d}+\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (7 a^{2}+8 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (19 a^{2}+16 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (33 a^{2}+32 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {4 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(265\) |
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Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.86 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {2 \, {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - 4 \, b^{2} - {\left ({\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (2 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.23 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, b^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 8 \, a b {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{4 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.57 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, {\left (3 \, a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {16 \, {\left (2 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2} + b^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 11.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{2}+b^2\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}+8\,b^2\right )-\frac {a^2}{2}+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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