Optimal. Leaf size=129 \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0947062, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 77} \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 77
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5 (a+x)}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-16 a^3+\frac{2 a^6}{(a-x)^3}-\frac{11 a^5}{(a-x)^2}+\frac{25 a^4}{a-x}-9 a^2 x-4 a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{25 a^4 \log (1-\sin (c+d x))}{d}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.469245, size = 83, normalized size = 0.64 \[ -\frac{a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+54 \sin ^2(c+d x)+192 \sin (c+d x)+\frac{120-132 \sin (c+d x)}{(\sin (c+d x)-1)^2}+300 \log (1-\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.1, size = 387, normalized size = 3. \begin{align*} -{\frac{5\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d}}-4\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}-5\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4\,d}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-3\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{25\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-25\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}+25\,{\frac{{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-6\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-12\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-25\,{\frac{{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10668, size = 147, normalized size = 1.14 \begin{align*} -\frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 54 \, a^{4} \sin \left (d x + c\right )^{2} + 300 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 192 \, a^{4} \sin \left (d x + c\right ) - \frac{12 \,{\left (11 \, a^{4} \sin \left (d x + c\right ) - 10 \, a^{4}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6719, size = 394, normalized size = 3.05 \begin{align*} -\frac{24 \, a^{4} \cos \left (d x + c\right )^{6} - 272 \, a^{4} \cos \left (d x + c\right )^{4} - 2393 \, a^{4} \cos \left (d x + c\right )^{2} + 1906 \, a^{4} + 2400 \,{\left (a^{4} \cos \left (d x + c\right )^{2} + 2 \, a^{4} \sin \left (d x + c\right ) - 2 \, a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \,{\left (8 \, a^{4} \cos \left (d x + c\right )^{4} - 96 \, a^{4} \cos \left (d x + c\right )^{2} + 181 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\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 [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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