Optimal. Leaf size=103 \[ \frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2834, 2611, 3770, 3473, 3475} \[ \frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 2834
Rule 3473
Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx &=a \int \sec (c+d x) \tan ^4(c+d x) \, dx+b \int \tan ^5(c+d x) \, dx\\ &=\frac {a \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d}-\frac {1}{4} (3 a) \int \sec (c+d x) \tan ^2(c+d x) \, dx-b \int \tan ^3(c+d x) \, dx\\ &=-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d}+\frac {1}{8} (3 a) \int \sec (c+d x) \, dx+b \int \tan (c+d x) \, dx\\ &=\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {b \log (\cos (c+d x))}{d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 106, normalized size = 1.03 \[ \frac {a \tan ^3(c+d x) \sec (c+d x)}{d}-\frac {a \left (6 \tan (c+d x) \sec ^3(c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )}{8 d}-\frac {b \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 104, normalized size = 1.01 \[ \frac {{\left (3 \, a - 8 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, a + 8 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 16 \, b \cos \left (d x + c\right )^{2} - 2 \, {\left (5 \, a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right ) + 4 \, b}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 100, normalized size = 0.97 \[ \frac {{\left (3 \, a - 8 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (3 \, a + 8 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (6 \, b \sin \left (d x + c\right )^{4} + 5 \, a \sin \left (d x + c\right )^{3} - 4 \, b \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 133, normalized size = 1.29 \[ \frac {a \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {3 a \sin \left (d x +c \right )}{8 d}+\frac {3 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 100, normalized size = 0.97 \[ \frac {{\left (3 \, a - 8 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, a + 8 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + \frac {2 \, {\left (5 \, a \sin \left (d x + c\right )^{3} + 8 \, b \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right ) - 6 \, b\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.03, size = 221, normalized size = 2.15 \[ \frac {b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-8\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {3\,a}{8}+b\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {3\,a}{8}-b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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