Optimal. Leaf size=33 \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0427132, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]
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)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{a+x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{a}{x^4}+\frac{1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0240431, size = 33, normalized size = 1. \[ -\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.034, size = 27, normalized size = 0.8 \begin{align*}{\frac{a}{d} \left ( -{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11486, size = 35, normalized size = 1.06 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54374, size = 92, normalized size = 2.79 \begin{align*} \frac{3 \, a \sin \left (d x + c\right ) + 2 \, a}{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 = 1.25737, size = 35, normalized size = 1.06 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]