Optimal. Leaf size=68 \[ -\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0933341, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2706
Rule 2607
Rule 30
Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^5(c+d x) \csc (c+d x) \, dx}{a}+\frac{\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc (c+d x)}{a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc ^5(c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.149153, size = 61, normalized size = 0.9 \[ \frac{\csc ^6(c+d x) (78 \sin (c+d x)-5 (7 \sin (3 (c+d x))-3 \sin (5 (c+d x))+5)-15 \cos (4 (c+d x)))}{240 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.095, size = 67, normalized size = 1. \begin{align*}{\frac{1}{da} \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{-1}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{2}{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 = 1.12848, size = 89, normalized size = 1.31 \begin{align*} \frac{30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41345, size = 248, normalized size = 3.65 \begin{align*} \frac{15 \, \cos \left (d x + c\right )^{4} - 15 \, \cos \left (d x + c\right )^{2} - 2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 5}{30 \,{\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a 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 = 1.4623, size = 89, normalized size = 1.31 \begin{align*} \frac{30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]