Optimal. Leaf size=65 \[ -\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.042355, antiderivative size = 65, 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 = {3012, 3767} \[ -\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 3767
Rubi steps
\begin{align*} \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}+\frac{1}{5} (4 a+5 b) \int \csc ^4(c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{(4 a+5 b) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{5 d}\\ &=-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.028352, size = 95, normalized size = 1.46 \[ -\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{2 b \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cot \left ( dx+c \right ) +b \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943166, size = 61, normalized size = 0.94 \begin{align*} -\frac{15 \,{\left (a + b\right )} \tan \left (d x + c\right )^{4} + 5 \,{\left (2 \, a + b\right )} \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61107, size = 208, normalized size = 3.2 \begin{align*} -\frac{2 \,{\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{5} - 5 \,{\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (a + b\right )} \cos \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \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 = 1.1379, size = 82, normalized size = 1.26 \begin{align*} -\frac{15 \, a \tan \left (d x + c\right )^{4} + 15 \, b \tan \left (d x + c\right )^{4} + 10 \, a \tan \left (d x + c\right )^{2} + 5 \, b \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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