Optimal. Leaf size=30 \[ -\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0568223, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ -\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^5 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{(a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.0234424, size = 20, normalized size = 0.67 \[ -\frac{a^3 (\csc (c+d x)+1)^4}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 49, normalized size = 1.6 \begin{align*}{\frac{{a}^{3}}{d} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}- \left ( \sin \left ( dx+c \right ) \right ) ^{-3}-{\frac{3}{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 = 1.01684, size = 73, normalized size = 2.43 \begin{align*} -\frac{4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.59412, size = 170, normalized size = 5.67 \begin{align*} \frac{6 \, a^{3} \cos \left (d x + c\right )^{2} - 7 \, a^{3} + 4 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\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 = 1.18985, size = 73, normalized size = 2.43 \begin{align*} -\frac{4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]