Optimal. Leaf size=126 \[ \frac{\sec ^4(c+d x) \left (a^2+2 a b \sin (c+d x)+b^2\right )}{4 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a (4 a+3 b) \log (1-\sin (c+d x))}{8 d}-\frac{a (4 a-3 b) \log (\sin (c+d x)+1)}{8 d}+\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.217093, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2837, 12, 1805, 823, 801} \[ \frac{\sec ^4(c+d x) \left (a^2+2 a b \sin (c+d x)+b^2\right )}{4 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a (4 a+3 b) \log (1-\sin (c+d x))}{8 d}-\frac{a (4 a-3 b) \log (\sin (c+d x)+1)}{8 d}+\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 1805
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{b (a+x)^2}{x \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^6 \operatorname{Subst}\left (\int \frac{(a+x)^2}{x \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}-\frac{b^4 \operatorname{Subst}\left (\int \frac{-4 a^2-6 a x}{x \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d}+\frac{\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^2 b^2-6 a b^2 x}{x \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d}+\frac{\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{a (4 a+3 b)}{b-x}-\frac{8 a^2}{x}+\frac{a (4 a-3 b)}{b+x}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=-\frac{a (4 a+3 b) \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a (4 a-3 b) \log (1+\sin (c+d x))}{8 d}+\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d}+\frac{\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.917146, size = 137, normalized size = 1.09 \[ \frac{16 a^2 \log (\sin (c+d x))-\frac{(a+b) (5 a+b)}{\sin (c+d x)-1}+\frac{(a-b) (5 a-b)}{\sin (c+d x)+1}+\frac{(a+b)^2}{(\sin (c+d x)-1)^2}+\frac{(a-b)^2}{(\sin (c+d x)+1)^2}-2 a (4 a+3 b) \log (1-\sin (c+d x))-2 a (4 a-3 b) \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 125, normalized size = 1. \begin{align*}{\frac{{a}^{2}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{ab\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{3\,ab\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{4\,d}}+{\frac{3\,ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{{b}^{2}}{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 = 0.982631, size = 176, normalized size = 1.4 \begin{align*} \frac{8 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) -{\left (4 \, a^{2} - 3 \, a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (4 \, a^{2} + 3 \, a b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (3 \, a b \sin \left (d x + c\right )^{3} + 2 \, a^{2} \sin \left (d x + c\right )^{2} - 5 \, a b \sin \left (d x + c\right ) - 3 \, a^{2} - b^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02736, size = 360, normalized size = 2.86 \begin{align*} \frac{8 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) -{\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, b^{2} + 2 \,{\left (3 \, a b \cos \left (d x + c\right )^{2} + 2 \, a b\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{4}} \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 = 1.23934, size = 181, normalized size = 1.44 \begin{align*} \frac{8 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) -{\left (4 \, a^{2} - 3 \, a b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) -{\left (4 \, a^{2} + 3 \, a b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (3 \, a^{2} \sin \left (d x + c\right )^{4} - 3 \, a b \sin \left (d x + c\right )^{3} - 8 \, a^{2} \sin \left (d x + c\right )^{2} + 5 \, a b \sin \left (d x + c\right ) + 6 \, a^{2} + b^{2}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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