Optimal. Leaf size=102 \[ \frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.142007, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2722, 2635, 8, 2592, 321, 206, 3473, 2565, 30} \[ \frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rule 2635
Rule 8
Rule 2592
Rule 321
Rule 206
Rule 3473
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \left (3 a b^2 \cos ^2(c+d x)+3 a^2 b \cos (c+d x) \cot (c+d x)+a^3 \cot ^2(c+d x)+b^3 \cos ^2(c+d x) \sin (c+d x)\right ) \, dx\\ &=a^3 \int \cot ^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \cos (c+d x) \cot (c+d x) \, dx+\left (3 a b^2\right ) \int \cos ^2(c+d x) \, dx+b^3 \int \cos ^2(c+d x) \sin (c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \cos (c+d x) \sin (c+d x)}{2 d}-a^3 \int 1 \, dx+\frac{1}{2} \left (3 a b^2\right ) \int 1 \, dx-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac{b^3 \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a^3 x+\frac{3}{2} a b^2 x+\frac{3 a^2 b \cos (c+d x)}{d}-\frac{b^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a^3 x+\frac{3}{2} a b^2 x-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 b \cos (c+d x)}{d}-\frac{b^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.2903, size = 143, normalized size = 1.4 \[ \frac{\left (36 a^2 b-3 b^3\right ) \cos (c+d x)+6 a \left (a^2 \tan \left (\frac{1}{2} (c+d x)\right )-2 a^2 c-2 a^2 d x+6 a b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-6 a b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3 b^2 c+3 b^2 d x\right )-6 a^3 \cot \left (\frac{1}{2} (c+d x)\right )+9 a b^2 \sin (2 (c+d x))-b^3 \cos (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.084, size = 125, normalized size = 1.2 \begin{align*} -{a}^{3}x-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}c}{d}}+3\,{\frac{{a}^{2}b\cos \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{2}b\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,a{b}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{3\,a{b}^{2}x}{2}}+{\frac{3\,a{b}^{2}c}{2\,d}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.62115, size = 128, normalized size = 1.25 \begin{align*} -\frac{4 \, b^{3} \cos \left (d x + c\right )^{3} + 12 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{3} - 9 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2} - 18 \, a^{2} b{\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5575, size = 370, normalized size = 3.63 \begin{align*} -\frac{9 \, a b^{2} \cos \left (d x + c\right )^{3} + 9 \, a^{2} b \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 9 \, a^{2} b \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right ) +{\left (2 \, b^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{2} b \cos \left (d x + c\right ) + 3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} d x\right )} \sin \left (d x + c\right )}{6 \, d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18499, size = 269, normalized size = 2.64 \begin{align*} \frac{18 \, a^{2} b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} - \frac{3 \,{\left (6 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{2 \,{\left (9 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 18 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 18 \, a^{2} b + 2 \, b^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]