Optimal. Leaf size=89 \[ \frac {35 \sec ^3(a+b x)}{24 b}+\frac {35 \sec (a+b x)}{8 b}-\frac {35 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2622, 288, 302, 207} \[ \frac {35 \sec ^3(a+b x)}{24 b}+\frac {35 \sec (a+b x)}{8 b}-\frac {35 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 288
Rule 302
Rule 2622
Rubi steps
\begin {align*} \int \csc ^5(a+b x) \sec ^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac {7 \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{4 b}\\ &=-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=\frac {35 \sec (a+b x)}{8 b}+\frac {35 \sec ^3(a+b x)}{24 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=-\frac {35 \tanh ^{-1}(\cos (a+b x))}{8 b}+\frac {35 \sec (a+b x)}{8 b}+\frac {35 \sec ^3(a+b x)}{24 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}\\ \end {align*}
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Mathematica [B] time = 0.44, size = 268, normalized size = 3.01 \[ -\frac {\csc ^{10}(a+b x) \left (658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (5 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (7 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+3 \cos (a+b x) \left (-105 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+105 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+76\right )+315 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (5 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (7 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-204\right )}{24 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 148, normalized size = 1.66 \[ \frac {210 \, \cos \left (b x + a\right )^{6} - 350 \, \cos \left (b x + a\right )^{4} + 112 \, \cos \left (b x + a\right )^{2} - 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 16}{48 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 209, normalized size = 2.35 \[ \frac {\frac {3 \, {\left (\frac {24 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {210 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {72 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {256 \, {\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {6 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 5\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 420 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{192 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 99, normalized size = 1.11 \[ -\frac {1}{4 b \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{3}}+\frac {7}{12 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{3}}-\frac {35}{24 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )}+\frac {35}{8 b \cos \left (b x +a \right )}+\frac {35 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 91, normalized size = 1.02 \[ \frac {\frac {2 \, {\left (105 \, \cos \left (b x + a\right )^{6} - 175 \, \cos \left (b x + a\right )^{4} + 56 \, \cos \left (b x + a\right )^{2} + 8\right )}}{\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}} - 105 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 105 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 78, normalized size = 0.88 \[ \frac {\frac {35\,{\cos \left (a+b\,x\right )}^6}{8}-\frac {175\,{\cos \left (a+b\,x\right )}^4}{24}+\frac {7\,{\cos \left (a+b\,x\right )}^2}{3}+\frac {1}{3}}{b\,\left ({\cos \left (a+b\,x\right )}^7-2\,{\cos \left (a+b\,x\right )}^5+{\cos \left (a+b\,x\right )}^3\right )}-\frac {35\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{8\,b} \]
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 \[ \int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin ^{5}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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