Optimal. Leaf size=66 \[ \frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {5 \csc (a+b x)}{16 b}-\frac {5 \csc ^3(a+b x)}{48 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4373, 2701,
294, 308, 213} \begin {gather*} -\frac {5 \csc ^3(a+b x)}{48 b}-\frac {5 \csc (a+b x)}{16 b}+\frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 294
Rule 308
Rule 2701
Rule 4373
Rubi steps
\begin {align*} \int \csc (a+b x) \csc ^3(2 a+2 b x) \, dx &=\frac {1}{8} \int \csc ^4(a+b x) \sec ^3(a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {5 \csc (a+b x)}{16 b}-\frac {5 \csc ^3(a+b x)}{48 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {5 \csc (a+b x)}{16 b}-\frac {5 \csc ^3(a+b x)}{48 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 31, normalized size = 0.47 \begin {gather*} -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\sin ^2(a+b x)\right )}{24 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 69, normalized size = 1.05
method | result | size |
default | \(\frac {-\frac {1}{3 \sin \left (x b +a \right )^{3} \cos \left (x b +a \right )^{2}}+\frac {5}{6 \sin \left (x b +a \right ) \cos \left (x b +a \right )^{2}}-\frac {5}{2 \sin \left (x b +a \right )}+\frac {5 \ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{2}}{8 b}\) | \(69\) |
risch | \(-\frac {i \left (15 \,{\mathrm e}^{9 i \left (x b +a \right )}-20 \,{\mathrm e}^{7 i \left (x b +a \right )}-22 \,{\mathrm e}^{5 i \left (x b +a \right )}-20 \,{\mathrm e}^{3 i \left (x b +a \right )}+15 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{16 b}+\frac {5 \ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{16 b}\) | \(126\) |
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. 1780 vs.
\(2 (58) = 116\).
time = 0.55, size = 1780, normalized size = 26.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs.
\(2 (58) = 116\).
time = 3.45, size = 130, normalized size = 1.97 \begin {gather*} -\frac {30 \, \cos \left (b x + a\right )^{4} - 15 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 15 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 40 \, \cos \left (b x + a\right )^{2} + 6}{96 \, {\left (b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )} \end {gather*}
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 \begin {gather*} \int \csc {\left (a + b x \right )} \csc ^{3}{\left (2 a + 2 b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 72, normalized size = 1.09 \begin {gather*} -\frac {\frac {6 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \frac {4 \, {\left (6 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 15 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 15 \, \log \left (-\sin \left (b x + a\right ) + 1\right )}{96 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 61, normalized size = 0.92 \begin {gather*} \frac {5\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}-\frac {-\frac {5\,{\sin \left (a+b\,x\right )}^4}{16}+\frac {5\,{\sin \left (a+b\,x\right )}^2}{24}+\frac {1}{24}}{b\,\left ({\sin \left (a+b\,x\right )}^3-{\sin \left (a+b\,x\right )}^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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