Integrand size = 27, antiderivative size = 48 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \text {arctanh}(\cos (c+d x))}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2700, 14, 2702, 327, 213} \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}-\frac {b \text {arctanh}(\cos (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Rule 14
Rule 213
Rule 327
Rule 2700
Rule 2702
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+b \int \csc (c+d x) \sec ^2(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {b \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {b \sec (c+d x)}{d}+\frac {a \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {b \text {arctanh}(\cos (c+d x))}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.42 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot (c+d x)}{d}-\frac {b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.68 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {a \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
default | \(\frac {a \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
parallelrisch | \(\frac {\left (2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-6 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 b}{2 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 d}\) | \(91\) |
risch | \(\frac {2 b \,{\mathrm e}^{3 i \left (d x +c \right )}-4 i a -2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(96\) |
norman | \(\frac {\frac {a}{2 d}-\frac {5 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {5 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(149\) |
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.00 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - b \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 4 \, a \cos \left (d x + c\right )^{2} - 2 \, b \sin \left (d x + c\right ) - 2 \, a}{2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
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\[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \csc ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {b {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, a {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (48) = 96\).
Time = 0.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.15 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{6 \, d} \]
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Time = 10.96 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.92 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-a}{d\,\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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