Optimal. Leaf size=112 \[ -\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}+\frac {105 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{256 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b} \]
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Rubi [A]
time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4373, 2702,
294, 308, 213} \begin {gather*} \frac {35 \sec ^3(a+b x)}{256 b}+\frac {105 \sec (a+b x)}{256 b}-\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 294
Rule 308
Rule 2702
Rule 4373
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac {1}{16} \int \csc ^7(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^{10}}{\left (-1+x^2\right )^4} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {3 \text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {21 \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{128 b}\\ &=-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=\frac {105 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{256 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=-\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}+\frac {105 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{256 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(278\) vs. \(2(112)=224\).
time = 0.87, size = 278, normalized size = 2.48 \begin {gather*} \frac {\csc ^{12}(a+b x) \left (1150-4752 \cos (2 (a+b x))+1600 \cos (3 (a+b x))+504 \cos (4 (a+b x))+1680 \cos (6 (a+b x))-600 \cos (7 (a+b x))-630 \cos (8 (a+b x))+200 \cos (9 (a+b x))+2520 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-945 \cos (7 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+315 \cos (9 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-30 \cos (a+b x) \left (40+63 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-63 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )-2520 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+945 \cos (7 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-315 \cos (9 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )}{3072 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 107, normalized size = 0.96
method | result | size |
default | \(\frac {-\frac {1}{6 \sin \left (x b +a \right )^{6} \cos \left (x b +a \right )^{3}}-\frac {3}{8 \sin \left (x b +a \right )^{4} \cos \left (x b +a \right )^{3}}+\frac {7}{8 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}-\frac {35}{16 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )}+\frac {105}{16 \cos \left (x b +a \right )}+\frac {105 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16}}{16 b}\) | \(107\) |
risch | \(\frac {315 \,{\mathrm e}^{17 i \left (x b +a \right )}-840 \,{\mathrm e}^{15 i \left (x b +a \right )}-252 \,{\mathrm e}^{13 i \left (x b +a \right )}+2376 \,{\mathrm e}^{11 i \left (x b +a \right )}-1150 \,{\mathrm e}^{9 i \left (x b +a \right )}+2376 \,{\mathrm e}^{7 i \left (x b +a \right )}-252 \,{\mathrm e}^{5 i \left (x b +a \right )}-840 \,{\mathrm e}^{3 i \left (x b +a \right )}+315 \,{\mathrm e}^{i \left (x b +a \right )}}{384 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{6} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{3}}-\frac {105 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{256 b}+\frac {105 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{256 b}\) | \(167\) |
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. 4268 vs.
\(2 (100) = 200\).
time = 0.47, size = 4268, normalized size = 38.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.21, size = 194, normalized size = 1.73 \begin {gather*} \frac {630 \, \cos \left (b x + a\right )^{8} - 1680 \, \cos \left (b x + a\right )^{6} + 1386 \, \cos \left (b x + a\right )^{4} - 288 \, \cos \left (b x + a\right )^{2} - 315 \, {\left (\cos \left (b x + a\right )^{9} - 3 \, \cos \left (b x + a\right )^{7} + 3 \, \cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 315 \, {\left (\cos \left (b x + a\right )^{9} - 3 \, \cos \left (b x + a\right )^{7} + 3 \, \cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 32}{1536 \, {\left (b \cos \left (b x + a\right )^{9} - 3 \, b \cos \left (b x + a\right )^{7} + 3 \, b \cos \left (b x + a\right )^{5} - b \cos \left (b x + a\right )^{3}\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 ^{3}{\left (a + b x \right )} \csc ^{4}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs.
\(2 (100) = 200\).
time = 0.46, size = 268, normalized size = 2.39 \begin {gather*} -\frac {\frac {285 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {21 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {\frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {225 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {2966 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {3513 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {660 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - 1}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}\right )}^{3}} - 1260 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{6144 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 100, normalized size = 0.89 \begin {gather*} \frac {-\frac {105\,{\cos \left (a+b\,x\right )}^8}{256}+\frac {35\,{\cos \left (a+b\,x\right )}^6}{32}-\frac {231\,{\cos \left (a+b\,x\right )}^4}{256}+\frac {3\,{\cos \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,\left (-{\cos \left (a+b\,x\right )}^9+3\,{\cos \left (a+b\,x\right )}^7-3\,{\cos \left (a+b\,x\right )}^5+{\cos \left (a+b\,x\right )}^3\right )}-\frac {105\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{256\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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