Optimal. Leaf size=66 \[ \frac {5 \sec ^3(a+b x)}{6 b}+\frac {5 \sec (a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2622, 288, 302, 207} \[ \frac {5 \sec ^3(a+b x)}{6 b}+\frac {5 \sec (a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 288
Rule 302
Rule 2622
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sec ^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b}\\ &=-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{2 b}\\ &=\frac {5 \sec (a+b x)}{2 b}+\frac {5 \sec ^3(a+b x)}{6 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b}\\ &=-\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}+\frac {5 \sec (a+b x)}{2 b}+\frac {5 \sec ^3(a+b x)}{6 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 205, normalized size = 3.11 \[ \frac {2 \csc ^8(a+b x) \left (-40 \cos (2 (a+b x))+13 \cos (3 (a+b x))-30 \cos (4 (a+b x))+13 \cos (5 (a+b x))+15 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+15 \cos (5 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-15 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-15 \cos (5 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+\cos (a+b x) \left (30 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-30 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-26\right )+22\right )}{3 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 112, normalized size = 1.70 \[ \frac {30 \, \cos \left (b x + a\right )^{4} - 20 \, \cos \left (b x + a\right )^{2} - 15 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 4}{12 \, {\left (b \cos \left (b x + a\right )^{5} - b \cos \left (b x + a\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 163, normalized size = 2.47 \[ -\frac {\frac {3 \, {\left (\frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {16 \, {\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 7\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} - 30 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 78, normalized size = 1.18 \[ \frac {1}{3 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{3}}-\frac {5}{6 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )}+\frac {5}{2 b \cos \left (b x +a \right )}+\frac {5 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 73, normalized size = 1.11 \[ \frac {\frac {2 \, {\left (15 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} - 2\right )}}{\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 60, normalized size = 0.91 \[ \frac {-\frac {5\,{\cos \left (a+b\,x\right )}^4}{2}+\frac {5\,{\cos \left (a+b\,x\right )}^2}{3}+\frac {1}{3}}{b\,\left ({\cos \left (a+b\,x\right )}^3-{\cos \left (a+b\,x\right )}^5\right )}-\frac {5\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]
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 \[ \int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin ^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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