Optimal. Leaf size=129 \[ \frac{\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac{\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^7(c+d x)}{7 d}-\frac{a b \csc ^6(c+d x)}{3 d}+\frac{a b \csc ^4(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{b^2 \csc (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.159315, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 948} \[ \frac{\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac{\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^7(c+d x)}{7 d}-\frac{a b \csc ^6(c+d x)}{3 d}+\frac{a b \csc ^4(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{b^2 \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^8 (a+x)^2 \left (b^2-x^2\right )^2}{x^8} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )^2}{x^8} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^3 \operatorname{Subst}\left (\int \left (\frac{a^2 b^4}{x^8}+\frac{2 a b^4}{x^7}+\frac{-2 a^2 b^2+b^4}{x^6}-\frac{4 a b^2}{x^5}+\frac{a^2-2 b^2}{x^4}+\frac{2 a}{x^3}+\frac{1}{x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{b^2 \csc (c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}+\frac{a b \csc ^4(c+d x)}{d}+\frac{\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac{a b \csc ^6(c+d x)}{3 d}-\frac{a^2 \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.181635, size = 104, normalized size = 0.81 \[ -\frac{\csc (c+d x) \left (21 \left (b^2-2 a^2\right ) \csc ^4(c+d x)+35 \left (a^2-2 b^2\right ) \csc ^2(c+d x)+15 a^2 \csc ^6(c+d x)+35 a b \csc ^5(c+d x)-105 a b \csc ^3(c+d x)+105 a b \csc (c+d x)+105 b^2\right )}{105 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.095, size = 218, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{35} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{b}^{2} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.990311, size = 143, normalized size = 1.11 \begin{align*} -\frac{105 \, b^{2} \sin \left (d x + c\right )^{6} + 105 \, a b \sin \left (d x + c\right )^{5} - 105 \, a b \sin \left (d x + c\right )^{3} + 35 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 35 \, a b \sin \left (d x + c\right ) - 21 \,{\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + 15 \, a^{2}}{105 \, d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65978, size = 362, normalized size = 2.81 \begin{align*} -\frac{105 \, b^{2} \cos \left (d x + c\right )^{6} - 35 \,{\left (a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 28 \,{\left (a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 8 \, a^{2} - 56 \, b^{2} - 35 \,{\left (3 \, 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 )}{105 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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 = 1.21549, size = 159, normalized size = 1.23 \begin{align*} -\frac{105 \, b^{2} \sin \left (d x + c\right )^{6} + 105 \, a b \sin \left (d x + c\right )^{5} + 35 \, a^{2} \sin \left (d x + c\right )^{4} - 70 \, b^{2} \sin \left (d x + c\right )^{4} - 105 \, a b \sin \left (d x + c\right )^{3} - 42 \, a^{2} \sin \left (d x + c\right )^{2} + 21 \, b^{2} \sin \left (d x + c\right )^{2} + 35 \, a b \sin \left (d x + c\right ) + 15 \, a^{2}}{105 \, d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]