Integrand size = 19, antiderivative size = 38 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2801, 3554, 8, 2670, 14} \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {a \tan (c+d x)}{d}-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \]
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Rule 8
Rule 14
Rule 2670
Rule 2801
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \int \left (a \tan ^2(c+d x)+b \sin (c+d x) \tan ^2(c+d x)\right ) \, dx \\ & = a \int \tan ^2(c+d x) \, dx+b \int \sin (c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {a \tan (c+d x)}{d}-a \int 1 \, dx-\frac {b \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -a x+\frac {a \tan (c+d x)}{d}-\frac {b \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \arctan (\tan (c+d x))}{d}+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.48 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\frac {b \cos \left (2 d x +2 c \right )+\left (-2 a x d +4 b \right ) \cos \left (d x +c \right )+2 a \sin \left (d x +c \right )+3 b}{2 d \cos \left (d x +c \right )}\) | \(54\) |
derivativedivides | \(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+b \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(59\) |
default | \(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+b \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(59\) |
risch | \(-a x +\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i a +2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(70\) |
norman | \(\frac {a x -\frac {4 b}{d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(89\) |
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a d x \cos \left (d x + c\right ) - b \cos \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - b}{d \cos \left (d x + c\right )} \]
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\[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a - b {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \]
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Time = 0.39 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {{\left (d x + c\right )} a + \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, b\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
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Time = 11.88 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-a\,x-\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \]
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