Optimal. Leaf size=65 \[ -\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0707239, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ -\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a+x)^3}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{(a+x)^3}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}+\frac{3 a^2}{x^3}+\frac{3 a}{x^2}+\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \csc (c+d x)}{d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc ^3(c+d x)}{3 d}+\frac{a^3 \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0191501, size = 57, normalized size = 0.88 \[ a^3 \left (-\frac{\csc ^3(c+d x)}{3 d}-\frac{3 \csc ^2(c+d x)}{2 d}-\frac{3 \csc (c+d x)}{d}+\frac{\log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 64, normalized size = 1. \begin{align*} -3\,{\frac{{a}^{3}}{d\sin \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3\,{a}^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0883, size = 78, normalized size = 1.2 \begin{align*} \frac{6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - \frac{18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70971, size = 221, normalized size = 3.4 \begin{align*} -\frac{18 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) - 20 \, a^{3} - 6 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{6 \,{\left (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.
[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 [A] time = 1.23809, size = 80, normalized size = 1.23 \begin{align*} \frac{6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]