Optimal. Leaf size=72 \[ \frac{\sin (a-c) \sec (b x+c)}{b}-\frac{3 \cos (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\sin (a+b x)}{b} \]
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Rubi [A] time = 0.0707118, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {4576, 4579, 2637, 3770, 2606, 8, 2611} \[ \frac{\sin (a-c) \sec (b x+c)}{b}-\frac{3 \cos (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4576
Rule 4579
Rule 2637
Rule 3770
Rule 2606
Rule 8
Rule 2611
Rubi steps
\begin{align*} \int \sin (a+b x) \tan ^3(c+b x) \, dx &=\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{\tanh ^{-1}(\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) \operatorname{Subst}(\int 1 \, dx,x,\sec (c+b x))}{b}+\int \cos (a+b x) \, dx\\ &=-\frac{3 \tanh ^{-1}(\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*}
Mathematica [A] time = 0.363034, size = 70, normalized size = 0.97 \[ \frac{\sec ^2(b x+c) (2 \sin (a-b x-2 c)+\sin (a+3 b x+2 c)+5 \sin (a+b x))-12 \cos (a-c) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{b x}{2}\right )+\sin (c)\right )}{4 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.115, size = 186, normalized size = 2.6 \begin{align*}{\frac{-{\frac{i}{2}}{{\rm e}^{i \left ( bx+a \right ) }}}{b}}+{\frac{{\frac{i}{2}}{{\rm e}^{-i \left ( bx+a \right ) }}}{b}}-{\frac{{\frac{i}{2}} \left ( 3\,{{\rm e}^{i \left ( 3\,bx+5\,a+2\,c \right ) }}-{{\rm e}^{i \left ( 3\,bx+3\,a+4\,c \right ) }}+{{\rm e}^{i \left ( bx+5\,a \right ) }}-3\,{{\rm e}^{i \left ( bx+3\,a+2\,c \right ) }} \right ) }{b \left ({{\rm e}^{2\,i \left ( bx+a+c \right ) }}+{{\rm e}^{2\,ia}} \right ) ^{2}}}+{\frac{3\,\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-i{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{2\,b}}-{\frac{3\,\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+i{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.97578, size = 1386, normalized size = 19.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.564859, size = 1015, normalized size = 14.1 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right ) \tan \left (b x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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