Optimal. Leaf size=74 \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \cot (c+d x)}{d}+a^3 x-\frac{b^2 (3 a+2 b) \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0462957, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4128, 390, 203} \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \cot (c+d x)}{d}+a^3 x-\frac{b^2 (3 a+2 b) \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 390
Rule 203
Rubi steps
\begin{align*} \int \left (a+b \csc ^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b+b x^2\right )^3}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (b \left (3 a^2+3 a b+b^2\right )+b^2 (3 a+2 b) x^2+b^3 x^4+\frac{a^3}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{b \left (3 a^2+3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a+2 b) \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=a^3 x-\frac{b \left (3 a^2+3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a+2 b) \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 3.5706, size = 112, normalized size = 1.51 \[ \frac{8 \sin ^6(c+d x) \left (a+b \csc ^2(c+d x)\right )^3 \left (b \cot (c+d x) \left (45 a^2+b (15 a+4 b) \csc ^2(c+d x)+30 a b+3 b^2 \csc ^4(c+d x)+8 b^2\right )-15 a^3 (c+d x)\right )}{15 d (a \cos (2 (c+d x))-a-2 b)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 83, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c \right ) -3\,\cot \left ( dx+c \right ){a}^{2}b+3\,a{b}^{2} \left ( -2/3-1/3\, \left ( \csc \left ( dx+c \right ) \right ) ^{2} \right ) \cot \left ( dx+c \right ) +{b}^{3} \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996448, size = 122, normalized size = 1.65 \begin{align*} a^{3} x - \frac{3 \, a^{2} b}{d \tan \left (d x + c\right )} - \frac{{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a b^{2}}{d \tan \left (d x + c\right )^{3}} - \frac{{\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} b^{3}}{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 [B] time = 0.487173, size = 386, normalized size = 5.22 \begin{align*} -\frac{{\left (45 \, a^{2} b + 30 \, a b^{2} + 8 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 5 \,{\left (18 \, a^{2} b + 15 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right ) - 15 \,{\left (a^{3} d x \cos \left (d x + c\right )^{4} - 2 \, a^{3} d x \cos \left (d x + c\right )^{2} + a^{3} d x\right )} \sin \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.56044, size = 285, normalized size = 3.85 \begin{align*} \frac{3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 25 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 480 \,{\left (d x + c\right )} a^{3} + 720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 540 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 150 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 540 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 150 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 25 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, b^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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