Optimal. Leaf size=65 \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0438446, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 43} \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2707
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^3}{x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a^2+\frac{a^3}{x}+3 a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0281867, size = 65, normalized size = 1. \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 62, normalized size = 1. \begin{align*}{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3} \left ( \sin \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 = 0.986767, size = 74, normalized size = 1.14 \begin{align*} \frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 18 \, a^{3} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75093, size = 146, normalized size = 2.25 \begin{align*} -\frac{9 \, a^{3} \cos \left (d x + c\right )^{2} - 6 \, a^{3} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 10 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \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 = 1.26631, size = 76, normalized size = 1.17 \begin{align*} \frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 18 \, a^{3} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]