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 = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 8
Rule 2606
Rule 2611
Rule 2637
Rule 3770
Rule 4576
Rule 4579
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*}
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Mathematica [A] time = 0.37, 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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fricas [B] time = 0.53, size = 376, normalized size = 5.22 \[ -\frac {\frac {3 \, \sqrt {2} {\left (2 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - 2 \, {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} + \cos \left (-2 \, a + 2 \, c\right )\right )} \cos \left (b x + a\right )^{2} + \cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 4 \, {\left (4 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 3 \, \cos \left (-2 \, a + 2 \, c\right ) + 5\right )} \sin \left (b x + a\right ) - 4 \, {\left (4 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )} \sin \left (-2 \, a + 2 \, c\right )}{8 \, {\left (2 \, b \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - b \cos \left (-2 \, a + 2 \, c\right ) + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (b x + a\right ) \tan \left (b x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.00, size = 186, normalized size = 2.58 \[ -\frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}+\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}-\frac {i \left (3 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-{\mathrm e}^{i \left (3 b x +3 a +4 c \right )}+{\mathrm e}^{i \left (b x +5 a \right )}-3 \,{\mathrm e}^{i \left (b x +3 a +2 c \right )}\right )}{2 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 1027, normalized size = 14.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\left (a + b x \right )} \tan ^{3}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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