Optimal. Leaf size=60 \[ -\frac {\cos (a-c) \cot (c+b x)}{b}-\frac {\cos (a-c) \cot ^3(c+b x)}{3 b}-\frac {\csc ^4(c+b x) \sin (a-c)}{4 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 60, 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 = {4678, 2686, 30,
3852} \begin {gather*} -\frac {\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c)}{b}-\frac {\sin (a-c) \csc ^4(b x+c)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 3852
Rule 4678
Rubi steps
\begin {align*} \int \csc ^5(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc ^4(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^4(c+b x) \, dx\\ &=-\frac {\cos (a-c) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+b x)\right )}{b}-\frac {\sin (a-c) \text {Subst}\left (\int x^3 \, dx,x,\csc (c+b x)\right )}{b}\\ &=-\frac {\cos (a-c) \cot (c+b x)}{b}-\frac {\cos (a-c) \cot ^3(c+b x)}{3 b}-\frac {\csc ^4(c+b x) \sin (a-c)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 58, normalized size = 0.97 \begin {gather*} \frac {(3 \cos (a)+\cos (a-c) (-4 \cos (c+2 b x)+\cos (3 c+4 b x))) \csc \left (\frac {c}{2}\right ) \csc ^4(c+b x) \sec \left (\frac {c}{2}\right )}{24 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. \(320\) vs.
\(2(56)=112\).
time = 2.85, size = 321, normalized size = 5.35
method | result | size |
risch | \(\frac {2 i \left (6 \,{\mathrm e}^{i \left (4 b x +9 a +3 c \right )}-4 \,{\mathrm e}^{i \left (2 b x +9 a +c \right )}-4 \,{\mathrm e}^{i \left (2 b x +7 a +3 c \right )}+{\mathrm e}^{i \left (9 a -c \right )}+{\mathrm e}^{i \left (7 a +c \right )}\right )}{3 \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{4} b}\) | \(97\) |
default | \(\frac {-\frac {1}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )}-\frac {3 \sin \left (a \right ) \cos \left (c \right )-3 \cos \left (a \right ) \sin \left (c \right )}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )^{2}}-\frac {\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )+3 \left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-4 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+3 \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 )}{3 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )^{3}}-\frac {\left (\sin \left (a \right ) \cos \left (c \right )-\cos \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 (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right )^{4}}}{b}\) | \(321\) |
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. 1076 vs.
\(2 (56) = 112\).
time = 0.32, size = 1076, normalized size = 17.93 \begin {gather*} -\frac {2 \, {\left ({\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 )} \cos \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 )} \cos \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 )} \cos \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 )} \cos \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 )} \cos \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 )} \cos \left (2 \, b x + 4 \, c\right ) + {\left (\sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\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 )} \sin \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 )} \sin \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 )} \sin \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 )} \sin \left (4 \, b x + a + 5 \, c\right ) - 4 \, \cos \left (a + c\right ) \sin \left (2 \, b x + 2 \, a + 2 \, 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 )} \sin \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 )} \sin \left (2 \, b x + 4 \, c\right ) - {\left (\cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 4 \, \cos \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 = 2.68, size = 75, normalized size = 1.25 \begin {gather*} \frac {4 \, {\left (2 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) + 3 \, \sin \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )}} \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. 301 vs.
\(2 (56) = 112\).
time = 0.42, size = 301, normalized size = 5.02 \begin {gather*} -\frac {6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} + 24 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (b x + c\right )^{3} + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 8 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right ) - 3 \, \tan \left (\frac {1}{2} \, c\right )}{6 \, {\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 \tan \left (b x + c\right )^{4}} \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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