Optimal. Leaf size=54 \[ \frac{a A \cos ^3(c+d x)}{3 d}-\frac{2 a A \cos (c+d x)}{d}+\frac{a A \sin (c+d x) \cos (c+d x)}{d}-a A x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0902227, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {21, 3788, 2635, 8, 4044, 3013} \[ \frac{a A \cos ^3(c+d x)}{3 d}-\frac{2 a A \cos (c+d x)}{d}+\frac{a A \sin (c+d x) \cos (c+d x)}{d}-a A x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 3788
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx &=\frac{A \int (a-a \csc (c+d x))^2 \sin ^3(c+d x) \, dx}{a}\\ &=\frac{A \int \left (a^2+a^2 \csc ^2(c+d x)\right ) \sin ^3(c+d x) \, dx}{a}-(2 a A) \int \sin ^2(c+d x) \, dx\\ &=\frac{a A \cos (c+d x) \sin (c+d x)}{d}+\frac{A \int \sin (c+d x) \left (a^2+a^2 \sin ^2(c+d x)\right ) \, dx}{a}-(a A) \int 1 \, dx\\ &=-a A x+\frac{a A \cos (c+d x) \sin (c+d x)}{d}-\frac{A \operatorname{Subst}\left (\int \left (2 a^2-a^2 x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-a A x-\frac{2 a A \cos (c+d x)}{d}+\frac{a A \cos ^3(c+d x)}{3 d}+\frac{a A \cos (c+d x) \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.109858, size = 44, normalized size = 0.81 \[ \frac{a A (6 (\sin (2 (c+d x))-2 (c+d x))-21 \cos (c+d x)+\cos (3 (c+d x)))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 62, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -{\frac{Aa \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}}-2\,Aa \left ( -1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) -Aa\cos \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00711, size = 81, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a - 3 \,{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 6 \, A a \cos \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.481526, size = 128, normalized size = 2.37 \begin{align*} \frac{A a \cos \left (d x + c\right )^{3} - 3 \, A a d x + 3 \, A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, A a \cos \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 13.726, size = 139, normalized size = 2.57 \begin{align*} \begin{cases} - \frac{A a x \cot ^{2}{\left (c + d x \right )}}{\csc ^{2}{\left (c + d x \right )}} - \frac{A a x}{\csc ^{2}{\left (c + d x \right )}} - \frac{2 A a \cot ^{3}{\left (c + d x \right )}}{3 d \csc ^{3}{\left (c + d x \right )}} - \frac{A a \cot{\left (c + d x \right )}}{d \csc{\left (c + d x \right )}} + \frac{A a \cot{\left (c + d x \right )}}{d \csc ^{2}{\left (c + d x \right )}} - \frac{A a \cot{\left (c + d x \right )}}{d \csc ^{3}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \left (- A \csc{\left (c \right )} + A\right ) \left (- a \csc{\left (c \right )} + a\right )}{\csc ^{3}{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.52714, size = 128, normalized size = 2.37 \begin{align*} -\frac{3 \,{\left (d x + c\right )} A a + \frac{2 \,{\left (3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, A a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]