Optimal. Leaf size=118 \[ -\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0766889, antiderivative size = 118, 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 = {2836, 12, 88} \[ -\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a (a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^6+\frac{a^7}{x}-3 a^5 x-3 a^4 x^2+3 a^3 x^3+3 a^2 x^4-a x^5-x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{a \log (\sin (c+d x))}{d}+\frac{a \sin (c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^4(c+d x)}{4 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^6(c+d x)}{6 d}-\frac{a \sin ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.138611, size = 106, normalized size = 0.9 \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \left (-2 \sin ^6(c+d x)+9 \sin ^4(c+d x)-18 \sin ^2(c+d x)+12 \log (\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 128, normalized size = 1.1 \begin{align*}{\frac{16\,a\sin \left ( dx+c \right ) }{35\,d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{7\,d}}+{\frac{6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{35\,d}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{35\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00251, size = 123, normalized size = 1.04 \begin{align*} -\frac{60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left (\sin \left (d x + c\right )\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54706, size = 263, normalized size = 2.23 \begin{align*} \frac{70 \, a \cos \left (d x + c\right )^{6} + 105 \, a \cos \left (d x + c\right )^{4} + 210 \, a \cos \left (d x + c\right )^{2} + 420 \, a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 12 \,{\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{420 \, 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 = 1.17745, size = 124, normalized size = 1.05 \begin{align*} -\frac{60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]