Optimal. Leaf size=43 \[ \frac {\tan ^2(a+b x)}{2 b}-\frac {\cot ^2(a+b x)}{2 b}+\frac {2 \log (\tan (a+b x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2620, 266, 43} \[ \frac {\tan ^2(a+b x)}{2 b}-\frac {\cot ^2(a+b x)}{2 b}+\frac {2 \log (\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2620
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sec ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^3} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^2} \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}+\frac {2}{x}\right ) \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=-\frac {\cot ^2(a+b x)}{2 b}+\frac {2 \log (\tan (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 61, normalized size = 1.42 \[ 8 \left (-\frac {\csc ^2(a+b x)}{16 b}+\frac {\sec ^2(a+b x)}{16 b}+\frac {\log (\sin (a+b x))}{4 b}-\frac {\log (\cos (a+b x))}{4 b}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 102, normalized size = 2.37 \[ \frac {2 \, \cos \left (b x + a\right )^{2} - 2 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 2 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 1}{2 \, {\left (b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 188, normalized size = 4.37 \[ -\frac {\frac {{\left (\frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - \frac {8 \, {\left (\frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 3\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{2}} - 8 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 16 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 48, normalized size = 1.12 \[ \frac {1}{2 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {1}{\sin \left (b x +a \right )^{2} b}+\frac {2 \ln \left (\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 64, normalized size = 1.49 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{4} - \sin \left (b x + a\right )^{2}} + 2 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 2 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 39, normalized size = 0.91 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{2\,b}-\frac {1}{2\,b\,{\mathrm {tan}\left (a+b\,x\right )}^2}+\frac {2\,\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b} \]
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 \[ \int \frac {\sec ^{3}{\left (a + b x \right )}}{\sin ^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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