Optimal. Leaf size=38 \[ \frac {\sin (a-c) \tan (b x+c)}{b}+\frac {\cos (a-c) \sec ^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4580, 2606, 30, 3767, 8} \[ \frac {\sin (a-c) \tan (b x+c)}{b}+\frac {\cos (a-c) \sec ^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 3767
Rule 4580
Rubi steps
\begin {align*} \int \sec ^3(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^2(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^2(c+b x) \, dx\\ &=\frac {\cos (a-c) \operatorname {Subst}(\int x \, dx,x,\sec (c+b x))}{b}-\frac {\sin (a-c) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+b x))}{b}\\ &=\frac {\cos (a-c) \sec ^2(c+b x)}{2 b}+\frac {\sin (a-c) \tan (c+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 34, normalized size = 0.89 \[ \frac {\sec (c) \sec ^2(b x+c) (\sin (a-c) \sin (2 b x+c)+\cos (a))}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 42, normalized size = 1.11 \[ -\frac {2 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.85, size = 174, normalized size = 4.58 \[ \frac {\tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + c\right )^{2} + 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) - 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.14, size = 150, normalized size = 3.95 \[ \frac {\frac {1}{\left (\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right ) \left (\sin \relax (a ) \cos \relax (c )-\cos \relax (a ) \sin \relax (c )\right ) \left (-\tan \left (b x +a \right ) \cos \relax (a ) \sin \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \cos \relax (c )+\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )}+\frac {-\cos \relax (a ) \cos \relax (c )-\sin \relax (a ) \sin \relax (c )}{2 \left (\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right ) \left (\sin \relax (a ) \cos \relax (c )-\cos \relax (a ) \sin \relax (c )\right ) \left (-\tan \left (b x +a \right ) \cos \relax (a ) \sin \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \cos \relax (c )+\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 391, normalized size = 10.29 \[ \frac {{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) + {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) + {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} + 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) + b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) + b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.03 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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