Optimal. Leaf size=49 \[ \frac {3 \sec (a+b x)}{2 b}-\frac {3 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2622, 288, 321, 207} \[ \frac {3 \sec (a+b x)}{2 b}-\frac {3 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 288
Rule 321
Rule 2622
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sec ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b}\\ &=\frac {3 \sec (a+b x)}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b}\\ &=-\frac {3 \tanh ^{-1}(\cos (a+b x))}{2 b}+\frac {3 \sec (a+b x)}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 143, normalized size = 2.92 \[ \frac {\csc ^4(a+b x) \left (-6 \cos (2 (a+b x))+2 \cos (3 (a+b x))+3 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-3 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+\cos (a+b x) \left (3 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-3 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-2\right )+2\right )}{2 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 96, normalized size = 1.96 \[ \frac {6 \, \cos \left (b x + a\right )^{2} - 3 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 4}{4 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 140, normalized size = 2.86 \[ \frac {\frac {\frac {14 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}} - \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 6 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 1.16 \[ -\frac {1}{2 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )}+\frac {3}{2 b \cos \left (b x +a \right )}+\frac {3 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 61, normalized size = 1.24 \[ \frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{2} - 2\right )}}{\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 49, normalized size = 1.00 \[ -\frac {3\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b}-\frac {\frac {3\,{\cos \left (a+b\,x\right )}^2}{2}-1}{b\,\left (\cos \left (a+b\,x\right )-{\cos \left (a+b\,x\right )}^3\right )} \]
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 \[ \int \frac {\sec ^{2}{\left (a + b x \right )}}{\sin ^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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