Optimal. Leaf size=105 \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.094418, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 88} \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^4}{a^4 (a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^4}{(a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{a^2}{4 (a-x)^3}-\frac{3 a}{4 (a-x)^2}+\frac{11}{16 (a-x)}+\frac{a}{8 (a+x)^2}-\frac{5}{16 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{11 a \log (1-\sin (c+d x))}{16 d}-\frac{5 a \log (1+\sin (c+d x))}{16 d}+\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.230451, size = 106, normalized size = 1.01 \[ \frac{a \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{a \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d}-\frac{a \left (6 \tan (c+d x) \sec ^3(c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 133, normalized size = 1.3 \begin{align*}{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d}}-{\frac{3\,a\sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01102, size = 116, normalized size = 1.1 \begin{align*} -\frac{5 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a \sin \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) - 6 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53321, size = 351, normalized size = 3.34 \begin{align*} \frac{10 \, a \cos \left (d x + c\right )^{2} - 5 \,{\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 11 \,{\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a \sin \left (d x + c\right ) + 2 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.49534, size = 126, normalized size = 1.2 \begin{align*} -\frac{10 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 22 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (5 \, a \sin \left (d x + c\right ) + 3 \, a\right )}}{\sin \left (d x + c\right ) + 1} - \frac{33 \, a \sin \left (d x + c\right )^{2} - 42 \, a \sin \left (d x + c\right ) + 13 \, a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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