Integrand size = 15, antiderivative size = 72 \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=-\frac {3 \text {arctanh}(\sin (c+b x)) \cos (a-c)}{2 b}+\frac {\sec (c+b x) \sin (a-c)}{b}+\frac {\sin (a+b x)}{b}+\frac {\cos (a-c) \sec (c+b x) \tan (c+b x)}{2 b} \]
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Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {4672, 4675, 2717, 3855, 2686, 8, 2691} \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=-\frac {3 \cos (a-c) \text {arctanh}(\sin (b x+c))}{2 b}+\frac {\sin (a-c) \sec (b x+c)}{b}+\frac {\cos (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac {\sin (a+b x)}{b} \]
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Rule 8
Rule 2686
Rule 2691
Rule 2717
Rule 3855
Rule 4672
Rule 4675
Rubi steps \begin{align*} \text {integral}& = \cos (a-c) \int \sec (c+b x) \tan ^2(c+b x) \, dx-\int \cos (a+b x) \tan ^2(c+b x) \, dx \\ & = \frac {\cos (a-c) \sec (c+b x) \tan (c+b x)}{2 b}-\frac {1}{2} \cos (a-c) \int \sec (c+b x) \, dx+\sin (a-c) \int \sec (c+b x) \tan (c+b x) \, dx-\int \sin (a+b x) \tan (c+b x) \, dx \\ & = -\frac {\text {arctanh}(\sin (c+b x)) \cos (a-c)}{2 b}+\frac {\cos (a-c) \sec (c+b x) \tan (c+b x)}{2 b}-\cos (a-c) \int \sec (c+b x) \, dx+\frac {\sin (a-c) \text {Subst}(\int 1 \, dx,x,\sec (c+b x))}{b}+\int \cos (a+b x) \, dx \\ & = -\frac {3 \text {arctanh}(\sin (c+b x)) \cos (a-c)}{2 b}+\frac {\sec (c+b x) \sin (a-c)}{b}+\frac {\sin (a+b x)}{b}+\frac {\cos (a-c) \sec (c+b x) \tan (c+b x)}{2 b} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=\frac {-12 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+\sec ^2(c+b x) (2 \sin (a-2 c-b x)+5 \sin (a+b x)+\sin (a+2 c+3 b x))}{4 b} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.58
method | result | size |
risch | \(-\frac {i {\mathrm e}^{i \left (x b +a \right )}}{2 b}+\frac {i {\mathrm e}^{-i \left (x b +a \right )}}{2 b}-\frac {i \left (3 \,{\mathrm e}^{i \left (3 x b +5 a +2 c \right )}-{\mathrm e}^{i \left (3 x b +3 a +4 c \right )}+{\mathrm e}^{i \left (x b +5 a \right )}-3 \,{\mathrm e}^{i \left (x b +3 a +2 c \right )}\right )}{2 b \left ({\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}\) | \(186\) |
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Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (68) = 136\).
Time = 0.28 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.22 \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=-\frac {\frac {3 \, \sqrt {2} {\left (2 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - 2 \, {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} + \cos \left (-2 \, a + 2 \, c\right )\right )} \cos \left (b x + a\right )^{2} + \cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 4 \, {\left (4 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 3 \, \cos \left (-2 \, a + 2 \, c\right ) + 5\right )} \sin \left (b x + a\right ) - 4 \, {\left (4 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )} \sin \left (-2 \, a + 2 \, c\right )}{8 \, {\left (2 \, b \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - b \cos \left (-2 \, a + 2 \, c\right ) + b\right )}} \]
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\[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=\int \sin {\left (a + b x \right )} \tan ^{3}{\left (b x + c \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (68) = 136\).
Time = 0.42 (sec) , antiderivative size = 1027, normalized size of antiderivative = 14.26 \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=\text {Too large to display} \]
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\[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=\int { \sin \left (b x + a\right ) \tan \left (b x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \sin (a+b x) \tan ^3(c+b x) \, dx=\text {Hanged} \]
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