Optimal. Leaf size=120 \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{a b \sin ^2(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{4 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140823, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2837, 12, 948} \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{a b \sin ^2(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{4 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^4 (a+x)^2 \left (b^2-x^2\right )^2}{x^4} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )^2}{x^4} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1-\frac{2 b^2}{a^2}\right )+\frac{a^2 b^4}{x^4}+\frac{2 a b^4}{x^3}+\frac{-2 a^2 b^2+b^4}{x^2}-\frac{4 a b^2}{x}+2 a x+x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}-\frac{4 a b \log (\sin (c+d x))}{d}+\frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac{a b \sin ^2(c+d x)}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.274211, size = 103, normalized size = 0.86 \[ \frac{3 \left (a^2-2 b^2\right ) \sin (c+d x)+\left (6 a^2-3 b^2\right ) \csc (c+d x)-a^2 \csc ^3(c+d x)+3 a b \sin ^2(c+d x)-3 a b \csc ^2(c+d x)-12 a b \log (\sin (c+d x))+b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.087, size = 255, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}+{\frac{8\,{a}^{2}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{4\,{a}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-2\,{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{ab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-{\frac{8\,{b}^{2}\sin \left ( dx+c \right ) }{3\,d}}-{\frac{{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{4\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.992184, size = 139, normalized size = 1.16 \begin{align*} \frac{b^{2} \sin \left (d x + c\right )^{3} + 3 \, a b \sin \left (d x + c\right )^{2} - 12 \, a b \log \left (\sin \left (d x + c\right )\right ) + 3 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right ) - \frac{3 \, a b \sin \left (d x + c\right ) - 3 \,{\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + a^{2}}{\sin \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77982, size = 382, normalized size = 3.18 \begin{align*} \frac{2 \, b^{2} \cos \left (d x + c\right )^{6} - 6 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 24 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 24 \,{\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 16 \, a^{2} + 16 \, b^{2} - 3 \,{\left (2 \, a b \cos \left (d x + c\right )^{4} - 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \sin \left (d x + c\right )}{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 = 1.19856, size = 171, normalized size = 1.42 \begin{align*} \frac{b^{2} \sin \left (d x + c\right )^{3} + 3 \, a b \sin \left (d x + c\right )^{2} - 12 \, a b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 3 \, a^{2} \sin \left (d x + c\right ) - 6 \, b^{2} \sin \left (d x + c\right ) + \frac{22 \, a b \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} - 3 \, a b \sin \left (d x + c\right ) - a^{2}}{\sin \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]