Optimal. Leaf size=60 \[ -\frac {\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c)}{b}-\frac {\sin (a-c) \csc ^4(b x+c)}{4 b} \]
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Rubi [A] time = 0.05, 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 = {4582, 2606, 30, 3767} \[ -\frac {\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c)}{b}-\frac {\sin (a-c) \csc ^4(b x+c)}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 3767
Rule 4582
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) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+b x)\right )}{b}-\frac {\sin (a-c) \operatorname {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 = 0.38, size = 58, normalized size = 0.97 \[ \frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \csc ^4(b x+c) (\cos (a-c) (\cos (4 b x+3 c)-4 \cos (2 b x+c))+3 \cos (a))}{24 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 75, normalized size = 1.25 \[ \frac {4 \, {\left (2 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) + 3 \, \sin \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.25, size = 301, normalized size = 5.02 \[ -\frac {6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} + 24 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (b x + c\right )^{3} + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 8 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right ) - 3 \, \tan \left (\frac {1}{2} \, c\right )}{6 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b \tan \left (b x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.64, size = 321, normalized size = 5.35 \[ \frac {\frac {\left (\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right ) \left (\left (\cos ^{2}\relax (a )\right ) \left (\cos ^{2}\relax (c )\right )+\left (\cos ^{2}\relax (a )\right ) \left (\sin ^{2}\relax (c )\right )+\left (\cos ^{2}\relax (c )\right ) \left (\sin ^{2}\relax (a )\right )+\left (\sin ^{2}\relax (a )\right ) \left (\sin ^{2}\relax (c )\right )\right )}{4 \left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{4} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )^{4}}-\frac {1}{\left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{4} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )}-\frac {-3 \cos \relax (a ) \sin \relax (c )+3 \sin \relax (a ) \cos \relax (c )}{2 \left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{4} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )^{2}}-\frac {\left (\cos ^{2}\relax (a )\right ) \left (\cos ^{2}\relax (c )\right )+3 \left (\cos ^{2}\relax (a )\right ) \left (\sin ^{2}\relax (c )\right )-4 \cos \relax (a ) \cos \relax (c ) \sin \relax (a ) \sin \relax (c )+3 \left (\cos ^{2}\relax (c )\right ) \left (\sin ^{2}\relax (a )\right )+\left (\sin ^{2}\relax (a )\right ) \left (\sin ^{2}\relax (c )\right )}{3 \left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{4} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 1076, normalized size = 17.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.02 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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