Optimal. Leaf size=171 \[ \frac{b \sec ^4(c+d x) \left (a b \left (\frac{a^2}{b^2}+3\right ) \sin (c+d x)+3 a^2+b^2\right )}{4 d}+\frac{a b \sec ^2(c+d x) \left (b \left (\frac{7 a^2}{b^2}+9\right ) \sin (c+d x)+12 a\right )}{8 d}+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.370669, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2837, 12, 1805, 1802} \[ \frac{b \sec ^4(c+d x) \left (a b \left (\frac{a^2}{b^2}+3\right ) \sin (c+d x)+3 a^2+b^2\right )}{4 d}+\frac{a b \sec ^2(c+d x) \left (b \left (\frac{7 a^2}{b^2}+9\right ) \sin (c+d x)+12 a\right )}{8 d}+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 1805
Rule 1802
Rubi steps
\begin{align*} \int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{b^2 (a+x)^3}{x^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^7 \operatorname{Subst}\left (\int \frac{(a+x)^3}{x^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac{a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}-\frac{b^5 \operatorname{Subst}\left (\int \frac{-4 a^3-12 a^2 x-3 a \left (3+\frac{a^2}{b^2}\right ) x^2}{x^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac{a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac{a b \sec ^2(c+d x) \left (12 a+\left (9+\frac{7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{8 a^3+24 a^2 x+a \left (9+\frac{7 a^2}{b^2}\right ) x^2}{x^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac{a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac{a b \sec ^2(c+d x) \left (12 a+\left (9+\frac{7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac{b^3 \operatorname{Subst}\left (\int \left (\frac{3 a (a+b) (5 a+3 b)}{2 b^3 (b-x)}+\frac{8 a^3}{b^2 x^2}+\frac{24 a^2}{b^2 x}+\frac{3 a (5 a-3 b) (a-b)}{2 b^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 a^2 b \log (\sin (c+d x))}{d}+\frac{3 a (5 a-3 b) (a-b) \log (1+\sin (c+d x))}{16 d}+\frac{b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac{a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac{a b \sec ^2(c+d x) \left (12 a+\left (9+\frac{7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}\\ \end{align*}
Mathematica [A] time = 1.24584, size = 161, normalized size = 0.94 \[ -\frac{-48 a^2 b \log (\sin (c+d x))+16 a^3 \csc (c+d x)+\frac{(a+b)^2 (7 a+b)}{\sin (c+d x)-1}+\frac{(a-b)^2 (7 a-b)}{\sin (c+d x)+1}-\frac{(a+b)^3}{(\sin (c+d x)-1)^2}+\frac{(a-b)^3}{(\sin (c+d x)+1)^2}+3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))-3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 221, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}}{4\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{5\,{a}^{3}}{8\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{15\,{a}^{3}}{8\,d\sin \left ( dx+c \right ) }}+{\frac{15\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,{a}^{2}b}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{2}b}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{a}^{2}b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,a{b}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{9\,a{b}^{2}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{8\,d}}+{\frac{9\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{b}^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01105, size = 254, normalized size = 1.49 \begin{align*} \frac{48 \, a^{2} b \log \left (\sin \left (d x + c\right )\right ) + 3 \,{\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (12 \, a^{2} b \sin \left (d x + c\right )^{3} + 3 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{4} + 8 \, a^{3} - 5 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{2} - 2 \,{\left (9 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{5} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18192, size = 560, normalized size = 3.27 \begin{align*} \frac{48 \, a^{2} b \cos \left (d x + c\right )^{4} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \,{\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 3 \,{\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 6 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 4 \, a^{3} + 12 \, a b^{2} + 2 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (6 \, a^{2} b \cos \left (d x + c\right )^{2} + 3 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4} \sin \left (d x + c\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 = 1.35581, size = 284, normalized size = 1.66 \begin{align*} \frac{48 \, a^{2} b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 3 \,{\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \,{\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{16 \,{\left (3 \, a^{2} b \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )} + \frac{2 \,{\left (18 \, a^{2} b \sin \left (d x + c\right )^{4} - 7 \, a^{3} \sin \left (d x + c\right )^{3} - 9 \, a b^{2} \sin \left (d x + c\right )^{3} - 48 \, a^{2} b \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 15 \, a b^{2} \sin \left (d x + c\right ) + 36 \, a^{2} b + 2 \, b^{3}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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