Optimal. Leaf size=76 \[ -\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{24 b}+\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, 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 = {2691, 3853,
3855} \begin {gather*} -\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\tan (a+b x) \sec ^5(a+b x)}{6 b}-\frac {\tan (a+b x) \sec ^3(a+b x)}{24 b}-\frac {\tan (a+b x) \sec (a+b x)}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \sec ^5(a+b x) \tan ^2(a+b x) \, dx &=\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b}-\frac {1}{6} \int \sec ^5(a+b x) \, dx\\ &=-\frac {\sec ^3(a+b x) \tan (a+b x)}{24 b}+\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b}-\frac {1}{8} \int \sec ^3(a+b x) \, dx\\ &=-\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{24 b}+\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b}-\frac {1}{16} \int \sec (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{24 b}+\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 76, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{24 b}+\frac {\sec ^5(a+b x) \tan (a+b x)}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 84, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{6 \cos \left (b x +a \right )^{6}}+\frac {\sin ^{3}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{4}}+\frac {\sin ^{3}\left (b x +a \right )}{16 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{16}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{16}}{b}\) | \(84\) |
default | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{6 \cos \left (b x +a \right )^{6}}+\frac {\sin ^{3}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{4}}+\frac {\sin ^{3}\left (b x +a \right )}{16 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{16}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{16}}{b}\) | \(84\) |
risch | \(\frac {i \left (3 \,{\mathrm e}^{11 i \left (b x +a \right )}+17 \,{\mathrm e}^{9 i \left (b x +a \right )}-114 \,{\mathrm e}^{7 i \left (b x +a \right )}+114 \,{\mathrm e}^{5 i \left (b x +a \right )}-17 \,{\mathrm e}^{3 i \left (b x +a \right )}-3 \,{\mathrm e}^{i \left (b x +a \right )}\right )}{24 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{16 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{16 b}\) | \(124\) |
norman | \(\frac {\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {47 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {13 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {13 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {47 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{6}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{16 b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{16 b}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 91, normalized size = 1.20 \begin {gather*} \frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{96 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 84, normalized size = 1.11 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{6} \log \left (\sin \left (b x + a\right ) + 1\right ) - 3 \, \cos \left (b x + a\right )^{6} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (3 \, \cos \left (b x + a\right )^{4} + 2 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{96 \, b \cos \left (b x + a\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 73, normalized size = 0.96 \begin {gather*} \frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{96 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.33, size = 177, normalized size = 2.33 \begin {gather*} \frac {\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{11}}{8}+\frac {47\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^9}{24}+\frac {13\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^7}{4}+\frac {13\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^5}{4}+\frac {47\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3}{24}+\frac {\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{8}}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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