Optimal. Leaf size=59 \[ \frac {\cos (a-c) \sec ^4(c+b x)}{4 b}+\frac {\sin (a-c) \tan (c+b x)}{b}+\frac {\sin (a-c) \tan ^3(c+b x)}{3 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4676, 2686, 30,
3852} \begin {gather*} \frac {\sin (a-c) \tan ^3(b x+c)}{3 b}+\frac {\sin (a-c) \tan (b x+c)}{b}+\frac {\cos (a-c) \sec ^4(b x+c)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 3852
Rule 4676
Rubi steps
\begin {align*} \int \sec ^5(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^4(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^4(c+b x) \, dx\\ &=\frac {\cos (a-c) \text {Subst}\left (\int x^3 \, dx,x,\sec (c+b x)\right )}{b}-\frac {\sin (a-c) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+b x)\right )}{b}\\ &=\frac {\cos (a-c) \sec ^4(c+b x)}{4 b}+\frac {\sin (a-c) \tan (c+b x)}{b}+\frac {\sin (a-c) \tan ^3(c+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 48, normalized size = 0.81 \begin {gather*} \frac {\sec (c) \sec ^4(c+b x) (3 \cos (a)+\sin (a-c) (4 \sin (c+2 b x)+\sin (3 c+4 b x)))}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(323\) vs.
\(2(55)=110\).
time = 2.68, size = 324, normalized size = 5.49
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{i \left (4 b x +9 a +3 c \right )}+\frac {8 \,{\mathrm e}^{i \left (2 b x +9 a +c \right )}}{3}-\frac {8 \,{\mathrm e}^{i \left (2 b x +7 a +3 c \right )}}{3}+\frac {2 \,{\mathrm e}^{i \left (9 a -c \right )}}{3}-\frac {2 \,{\mathrm e}^{i \left (7 a +c \right )}}{3}}{\left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{4} b}\) | \(96\) |
default | \(\frac {-\frac {1}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{4} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}+\frac {3 \cos \left (a \right ) \cos \left (c \right )+3 \sin \left (a \right ) \sin \left (c \right )}{2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{4} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )+3 \left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )+4 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+3 \left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )+\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}{3 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{4} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{3}}+\frac {\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \left (\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )+\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )+\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )+\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )\right )}{4 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{4} \left (-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4}}}{b}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1074 vs.
\(2 (55) = 110\).
time = 0.29, size = 1074, normalized size = 18.20 \begin {gather*} \frac {2 \, {\left ({\left (6 \, \cos \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \cos \left (2 \, b x + 4 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (8 \, b x + a + 9 \, c\right ) + 4 \, {\left (6 \, \cos \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \cos \left (2 \, b x + 4 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (6 \, b x + a + 7 \, c\right ) + 6 \, {\left (4 \, \cos \left (2 \, b x + a + 3 \, c\right ) + \cos \left (a + c\right )\right )} \cos \left (4 \, b x + 2 \, a + 4 \, c\right ) + 6 \, {\left (6 \, \cos \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \cos \left (2 \, b x + 4 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 4 \, {\left (4 \, \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 ) - 4 \, {\left (4 \, \cos \left (2 \, b x + a + 3 \, c\right ) + \cos \left (a + c\right )\right )} \cos \left (2 \, b x + 4 \, c\right ) + {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + 4 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + {\left (6 \, \sin \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \sin \left (2 \, b x + 4 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (8 \, b x + a + 9 \, c\right ) + 4 \, {\left (6 \, \sin \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \sin \left (2 \, b x + 4 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (6 \, b x + a + 7 \, c\right ) + 6 \, {\left (4 \, \sin \left (2 \, b x + a + 3 \, c\right ) + \sin \left (a + c\right )\right )} \sin \left (4 \, b x + 2 \, a + 4 \, c\right ) + 6 \, {\left (6 \, \sin \left (4 \, b x + 2 \, a + 4 \, c\right ) + 4 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - 4 \, \sin \left (2 \, b x + 4 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 4 \, {\left (4 \, \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 ) - 4 \, {\left (4 \, \sin \left (2 \, b x + a + 3 \, c\right ) + \sin \left (a + c\right )\right )} \sin \left (2 \, b x + 4 \, c\right ) + {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 4 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )\right )}}{3 \, {\left (b \cos \left (8 \, b x + a + 9 \, c\right )^{2} + 16 \, b \cos \left (6 \, b x + a + 7 \, c\right )^{2} + 36 \, b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 16 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} + 8 \, 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 (8 \, b x + a + 9 \, c\right )^{2} + 16 \, b \sin \left (6 \, b x + a + 7 \, c\right )^{2} + 36 \, b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 16 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} + 8 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} + 2 \, {\left (4 \, b \cos \left (6 \, b x + a + 7 \, c\right ) + 6 \, b \cos \left (4 \, b x + a + 5 \, c\right ) + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) + b \cos \left (a + c\right )\right )} \cos \left (8 \, b x + a + 9 \, c\right ) + 8 \, {\left (6 \, b \cos \left (4 \, b x + a + 5 \, c\right ) + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) + b \cos \left (a + c\right )\right )} \cos \left (6 \, b x + a + 7 \, c\right ) + 12 \, {\left (4 \, 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 (4 \, b \sin \left (6 \, b x + a + 7 \, c\right ) + 6 \, b \sin \left (4 \, b x + a + 5 \, c\right ) + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) + b \sin \left (a + c\right )\right )} \sin \left (8 \, b x + a + 9 \, c\right ) + 8 \, {\left (6 \, b \sin \left (4 \, b x + a + 5 \, c\right ) + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) + b \sin \left (a + c\right )\right )} \sin \left (6 \, b x + a + 7 \, c\right ) + 12 \, {\left (4 \, 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 )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.02, size = 53, normalized size = 0.90 \begin {gather*} -\frac {4 \, {\left (2 \, \cos \left (b x + c\right )^{3} + \cos \left (b x + c\right )\right )} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 3 \, \cos \left (-a + c\right )}{12 \, b \cos \left (b x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs.
\(2 (55) = 110\).
time = 0.43, size = 327, normalized size = 5.54 \begin {gather*} \frac {3 \, \tan \left (b x + c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (b x + c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} + 12 \, \tan \left (b x + c\right )^{4} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 8 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 3 \, \tan \left (b x + c\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (b x + c\right )^{4} + 8 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} - 8 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, c\right ) + 24 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 24 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (b x + c\right )^{2} + 24 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) - 24 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )}{12 \, {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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