Optimal. Leaf size=274 \[ \frac{b^8 \log (a+b \sin (c+d x))}{a^3 d \left (a^2-b^2\right )^3}-\frac{\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{\left (24 a^2-57 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{b \csc (c+d x)}{a^2 d}+\frac{9 a+11 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{9 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.473842, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ \frac{b^8 \log (a+b \sin (c+d x))}{a^3 d \left (a^2-b^2\right )^3}-\frac{\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{\left (24 a^2-57 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{b \csc (c+d x)}{a^2 d}+\frac{9 a+11 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{9 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{b^3}{x^3 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^8 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^8 \operatorname{Subst}\left (\int \left (\frac{1}{8 b^6 (a+b) (b-x)^3}+\frac{9 a+11 b}{16 b^7 (a+b)^2 (b-x)^2}+\frac{24 a^2+57 a b+35 b^2}{16 b^8 (a+b)^3 (b-x)}+\frac{1}{a b^6 x^3}-\frac{1}{a^2 b^6 x^2}+\frac{3 a^2+b^2}{a^3 b^8 x}+\frac{1}{a^3 (a-b)^3 (a+b)^3 (a+x)}+\frac{1}{8 b^6 (-a+b) (b+x)^3}+\frac{-9 a+11 b}{16 (a-b)^2 b^7 (b+x)^2}+\frac{24 a^2-57 a b+35 b^2}{16 b^8 (-a+b)^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}-\frac{\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac{\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}+\frac{b^8 \log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^3 d}+\frac{1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac{9 a+11 b}{16 (a+b)^2 d (1-\sin (c+d x))}+\frac{1}{16 (a-b) d (1+\sin (c+d x))^2}+\frac{9 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.24281, size = 281, normalized size = 1.03 \[ \frac{b^8 \left (\frac{\csc (c+d x)}{a^2 b^7}-\frac{\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 b^8 (a+b)^3}+\frac{\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 b^8}-\frac{\left (24 a^2-57 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 b^8 (a-b)^3}+\frac{\log (a+b \sin (c+d x))}{a^3 (a-b)^3 (a+b)^3}+\frac{9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))}+\frac{9 a-11 b}{16 b^7 (a-b)^2 (b \sin (c+d x)+b)}+\frac{1}{16 b^6 (a+b) (b-b \sin (c+d x))^2}+\frac{1}{16 b^6 (a-b) (b \sin (c+d x)+b)^2}-\frac{\csc ^2(c+d x)}{2 a b^8}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.111, size = 371, normalized size = 1.4 \begin{align*}{\frac{{b}^{8}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}{a}^{3}}}+{\frac{1}{2\,d \left ( 8\,a+8\,b \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{9\,a}{16\,d \left ( a+b \right ) ^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{11\,b}{16\,d \left ( a+b \right ) ^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ){a}^{2}}{2\,d \left ( a+b \right ) ^{3}}}-{\frac{57\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) ab}{16\,d \left ( a+b \right ) ^{3}}}-{\frac{35\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ){b}^{2}}{16\,d \left ( a+b \right ) ^{3}}}+{\frac{1}{2\,d \left ( 8\,a-8\,b \right ) \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{9\,a}{16\,d \left ( a-b \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{11\,b}{16\,d \left ( a-b \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ){a}^{2}}{2\,d \left ( a-b \right ) ^{3}}}+{\frac{57\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) ab}{16\,d \left ( a-b \right ) ^{3}}}-{\frac{35\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ){b}^{2}}{16\,d \left ( a-b \right ) ^{3}}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}+{\frac{{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{b}{d{a}^{2}\sin \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07604, size = 570, normalized size = 2.08 \begin{align*} \frac{\frac{16 \, b^{8} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}} - \frac{{\left (24 \, a^{2} - 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (24 \, a^{2} + 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{2 \,{\left ({\left (15 \, a^{4} b - 27 \, a^{2} b^{3} + 8 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - 4 \, a^{5} + 8 \, a^{3} b^{2} - 4 \, a b^{4} - 4 \,{\left (3 \, a^{5} - 5 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} -{\left (25 \, a^{4} b - 45 \, a^{2} b^{3} + 16 \, b^{5}\right )} \sin \left (d x + c\right )^{3} + 2 \,{\left (9 \, a^{5} - 15 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \sin \left (d x + c\right )^{2} + 8 \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{6} - 2 \,{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{4} +{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}} + \frac{16 \,{\left (3 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 21.0347, size = 1418, normalized size = 5.18 \begin{align*} -\frac{4 \, a^{8} - 8 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - 8 \,{\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 5 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} - 16 \,{\left (b^{8} \cos \left (d x + c\right )^{6} - b^{8} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 16 \,{\left ({\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{6} -{\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) +{\left ({\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} -{\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left ({\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} -{\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \, a^{7} b - 4 \, a^{5} b^{3} + 2 \, a^{3} b^{5} -{\left (15 \, a^{7} b - 42 \, a^{5} b^{3} + 35 \, a^{3} b^{5} - 8 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} +{\left (5 \, a^{7} b - 14 \, a^{5} b^{3} + 9 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{6} -{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{4}\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 [B] time = 1.28829, size = 795, normalized size = 2.9 \begin{align*} \frac{\frac{16 \, b^{9} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}} - \frac{{\left (24 \, a^{2} - 57 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (24 \, a^{2} + 57 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{16 \,{\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac{2 \,{\left (4 \, b^{8} \sin \left (d x + c\right )^{6} + 15 \, a^{7} b \sin \left (d x + c\right )^{5} - 42 \, a^{5} b^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{5} \sin \left (d x + c\right )^{5} - 8 \, a b^{7} \sin \left (d x + c\right )^{5} - 12 \, a^{8} \sin \left (d x + c\right )^{4} + 32 \, a^{6} b^{2} \sin \left (d x + c\right )^{4} - 24 \, a^{4} b^{4} \sin \left (d x + c\right )^{4} + 4 \, a^{2} b^{6} \sin \left (d x + c\right )^{4} - 8 \, b^{8} \sin \left (d x + c\right )^{4} - 25 \, a^{7} b \sin \left (d x + c\right )^{3} + 70 \, a^{5} b^{3} \sin \left (d x + c\right )^{3} - 61 \, a^{3} b^{5} \sin \left (d x + c\right )^{3} + 16 \, a b^{7} \sin \left (d x + c\right )^{3} + 18 \, a^{8} \sin \left (d x + c\right )^{2} - 48 \, a^{6} b^{2} \sin \left (d x + c\right )^{2} + 38 \, a^{4} b^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{2} b^{6} \sin \left (d x + c\right )^{2} + 4 \, b^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{7} b \sin \left (d x + c\right ) - 24 \, a^{5} b^{3} \sin \left (d x + c\right ) + 24 \, a^{3} b^{5} \sin \left (d x + c\right ) - 8 \, a b^{7} \sin \left (d x + c\right ) - 4 \, a^{8} + 12 \, a^{6} b^{2} - 12 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right )}}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )}{\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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