Optimal. Leaf size=26 \[ \frac {35}{8} \tanh ^{-1}(\sin (x))-\frac {29}{8} \sec (x) \tan (x)+\frac {1}{4} \sec ^3(x) \tan (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4449, 1171, 393,
212} \begin {gather*} \frac {35}{8} \tanh ^{-1}(\sin (x))+\frac {1}{4} \tan (x) \sec ^3(x)-\frac {29}{8} \tan (x) \sec (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 1171
Rule 4449
Rubi steps
\begin {align*} \int \cos (4 x) \sec ^5(x) \, dx &=\text {Subst}\left (\int \frac {1-8 x^2+8 x^4}{\left (1-x^2\right )^3} \, dx,x,\sin (x)\right )\\ &=\frac {1}{4} \sec ^3(x) \tan (x)-\frac {1}{4} \text {Subst}\left (\int \frac {-3+32 x^2}{\left (1-x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=-\frac {29}{8} \sec (x) \tan (x)+\frac {1}{4} \sec ^3(x) \tan (x)+\frac {35}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )\\ &=\frac {35}{8} \tanh ^{-1}(\sin (x))-\frac {29}{8} \sec (x) \tan (x)+\frac {1}{4} \sec ^3(x) \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{8} \left (35 \tanh ^{-1}(\sin (x))-27 \sec ^3(x) \tan (x)+29 \sec (x) \tan ^3(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 31, normalized size = 1.19
method | result | size |
default | \(-\left (-\frac {\left (\sec ^{3}\left (x \right )\right )}{4}-\frac {3 \sec \left (x \right )}{8}\right ) \tan \left (x \right )+\frac {35 \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{8}-4 \sec \left (x \right ) \tan \left (x \right )\) | \(31\) |
risch | \(\frac {i \left (29 \,{\mathrm e}^{7 i x}+21 \,{\mathrm e}^{5 i x}-21 \,{\mathrm e}^{3 i x}-29 \,{\mathrm e}^{i x}\right )}{4 \left ({\mathrm e}^{2 i x}+1\right )^{4}}-\frac {35 \ln \left ({\mathrm e}^{i x}-i\right )}{8}+\frac {35 \ln \left ({\mathrm e}^{i x}+i\right )}{8}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (20) = 40\).
time = 2.32, size = 54, normalized size = 2.08 \begin {gather*} \frac {5 \, \sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{8 \, {\left (\sin \left (x\right )^{4} - 2 \, \sin \left (x\right )^{2} + 1\right )}} + \frac {3 \, \sin \left (x\right )}{\sin \left (x\right )^{2} - 1} + \frac {35}{16} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {35}{16} \, \log \left (\sin \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (20) = 40\).
time = 1.61, size = 43, normalized size = 1.65 \begin {gather*} \frac {35 \, \cos \left (x\right )^{4} \log \left (\sin \left (x\right ) + 1\right ) - 35 \, \cos \left (x\right )^{4} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (29 \, \cos \left (x\right )^{2} - 2\right )} \sin \left (x\right )}{16 \, \cos \left (x\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (27) = 54\).
time = 10.06, size = 75, normalized size = 2.88 \begin {gather*} - \frac {35 \log {\left (\sin {\left (x \right )} - 1 \right )}}{16} + \frac {35 \log {\left (\sin {\left (x \right )} + 1 \right )}}{16} - \frac {3 \sin ^{3}{\left (x \right )}}{8 \sin ^{4}{\left (x \right )} - 16 \sin ^{2}{\left (x \right )} + 8} + \frac {5 \sin {\left (x \right )}}{8 \sin ^{4}{\left (x \right )} - 16 \sin ^{2}{\left (x \right )} + 8} + \frac {8 \sin {\left (x \right )}}{2 \sin ^{2}{\left (x \right )} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 38, normalized size = 1.46 \begin {gather*} \frac {29 \, \sin \left (x\right )^{3} - 27 \, \sin \left (x\right )}{8 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac {35}{16} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {35}{16} \, \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 33, normalized size = 1.27 \begin {gather*} \frac {35\,\mathrm {atanh}\left (\sin \left (x\right )\right )}{8}-\frac {\frac {27\,\sin \left (x\right )}{8}-\frac {29\,{\sin \left (x\right )}^3}{8}}{{\sin \left (x\right )}^4-2\,{\sin \left (x\right )}^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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