Integrand size = 31, antiderivative size = 54 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=-a A x-\frac {2 a A \cos (c+d x)}{d}+\frac {a A \cos ^3(c+d x)}{3 d}+\frac {a A \cos (c+d x) \sin (c+d x)}{d} \]
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=\frac {a A (-21 \cos (c+d x)+\cos (3 (c+d x))+6 (-2 (c+d x)+\sin (2 (c+d x))))}{12 d} \]
Time = 0.45 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2011, 3042, 4275, 3042, 3115, 24, 4532, 3042, 3492, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx\) |
\(\Big \downarrow \) 2011 |
\(\displaystyle \frac {A \int (a-a \csc (c+d x))^2 \sin ^3(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \int \frac {(a-a \csc (c+d x))^2}{\csc (c+d x)^3}dx}{a}\) |
\(\Big \downarrow \) 4275 |
\(\displaystyle \frac {A \left (\int \left (\csc ^2(c+d x) a^2+a^2\right ) \sin ^3(c+d x)dx-2 a^2 \int \sin ^2(c+d x)dx\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \left (\int \frac {\csc (c+d x)^2 a^2+a^2}{\csc (c+d x)^3}dx-2 a^2 \int \sin (c+d x)^2dx\right )}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {A \left (\int \frac {\csc (c+d x)^2 a^2+a^2}{\csc (c+d x)^3}dx-2 a^2 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {A \left (\int \frac {\csc (c+d x)^2 a^2+a^2}{\csc (c+d x)^3}dx-2 a^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
\(\Big \downarrow \) 4532 |
\(\displaystyle \frac {A \left (\int \sin (c+d x) \left (\sin ^2(c+d x) a^2+a^2\right )dx-2 a^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \left (\int \sin (c+d x) \left (\sin (c+d x)^2 a^2+a^2\right )dx-2 a^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
\(\Big \downarrow \) 3492 |
\(\displaystyle \frac {A \left (-\frac {\int \left (2 a^2-a^2 \cos ^2(c+d x)\right )d\cos (c+d x)}{d}-2 a^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {A \left (-\frac {2 a^2 \cos (c+d x)-\frac {1}{3} a^2 \cos ^3(c+d x)}{d}-2 a^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )}{a}\) |
(A*(-((2*a^2*Cos[c + d*x] - (a^2*Cos[c + d*x]^3)/3)/d) - 2*a^2*(x/2 - (Cos [c + d*x]*Sin[c + d*x])/(2*d))))/a
3.1.24.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[2*a*(b/d) Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
Time = 0.76 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(-\frac {A a \left (12 d x +21 \cos \left (d x +c \right )-\cos \left (3 d x +3 c \right )-6 \sin \left (2 d x +2 c \right )+20\right )}{12 d}\) | \(44\) |
risch | \(-a A x -\frac {7 a A \cos \left (d x +c \right )}{4 d}+\frac {A a \cos \left (3 d x +3 c \right )}{12 d}+\frac {A a \sin \left (2 d x +2 c \right )}{2 d}\) | \(52\) |
derivativedivides | \(\frac {-A a \cos \left (d x +c \right )-2 A a \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {A a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(62\) |
default | \(\frac {-A a \cos \left (d x +c \right )-2 A a \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {A a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(62\) |
parts | \(-\frac {A a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}-\frac {a A \cos \left (d x +c \right )}{d}-\frac {2 A a \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(67\) |
norman | \(\frac {-\frac {2 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {8 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {10 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d}+\frac {2 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {2 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(179\) |
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=\frac {A a \cos \left (d x + c\right )^{3} - 3 \, A a d x + 3 \, A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, A a \cos \left (d x + c\right )}{3 \, d} \]
1/3*(A*a*cos(d*x + c)^3 - 3*A*a*d*x + 3*A*a*cos(d*x + c)*sin(d*x + c) - 6* A*a*cos(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (51) = 102\).
Time = 6.51 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.57 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=\begin {cases} - \frac {A a x \cot ^{2}{\left (c + d x \right )}}{\csc ^{2}{\left (c + d x \right )}} - \frac {A a x}{\csc ^{2}{\left (c + d x \right )}} - \frac {2 A a \cot ^{3}{\left (c + d x \right )}}{3 d \csc ^{3}{\left (c + d x \right )}} - \frac {A a \cot {\left (c + d x \right )}}{d \csc {\left (c + d x \right )}} + \frac {A a \cot {\left (c + d x \right )}}{d \csc ^{2}{\left (c + d x \right )}} - \frac {A a \cot {\left (c + d x \right )}}{d \csc ^{3}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \left (- A \csc {\left (c \right )} + A\right ) \left (- a \csc {\left (c \right )} + a\right )}{\csc ^{3}{\left (c \right )}} & \text {otherwise} \end {cases} \]
Piecewise((-A*a*x*cot(c + d*x)**2/csc(c + d*x)**2 - A*a*x/csc(c + d*x)**2 - 2*A*a*cot(c + d*x)**3/(3*d*csc(c + d*x)**3) - A*a*cot(c + d*x)/(d*csc(c + d*x)) + A*a*cot(c + d*x)/(d*csc(c + d*x)**2) - A*a*cot(c + d*x)/(d*csc(c + d*x)**3), Ne(d, 0)), (x*(-A*csc(c) + A)*(-a*csc(c) + a)/csc(c)**3, True ))
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 6 \, A a \cos \left (d x + c\right )}{6 \, d} \]
1/6*(2*(cos(d*x + c)^3 - 3*cos(d*x + c))*A*a - 3*(2*d*x + 2*c - sin(2*d*x + 2*c))*A*a - 6*A*a*cos(d*x + c))/d
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.76 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=-\frac {3 \, {\left (d x + c\right )} A a + \frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, A a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
-1/3*(3*(d*x + c)*A*a + 2*(3*A*a*tan(1/2*d*x + 1/2*c)^5 + 3*A*a*tan(1/2*d* x + 1/2*c)^4 + 12*A*a*tan(1/2*d*x + 1/2*c)^2 - 3*A*a*tan(1/2*d*x + 1/2*c) + 5*A*a)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 19.67 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.83 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx=\frac {-2\,A\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (A\,a\,\left (3\,c+3\,d\,x\right )-\frac {A\,a\,\left (9\,c+9\,d\,x+6\right )}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (A\,a\,\left (3\,c+3\,d\,x\right )-\frac {A\,a\,\left (9\,c+9\,d\,x+24\right )}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,A\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {A\,a\,\left (3\,c+3\,d\,x\right )}{3}-\frac {A\,a\,\left (3\,c+3\,d\,x+10\right )}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-A\,a\,x \]
(tan(c/2 + (d*x)/2)^4*(A*a*(3*c + 3*d*x) - (A*a*(9*c + 9*d*x + 6))/3) + ta n(c/2 + (d*x)/2)^2*(A*a*(3*c + 3*d*x) - (A*a*(9*c + 9*d*x + 24))/3) - 2*A* a*tan(c/2 + (d*x)/2)^5 + (A*a*(3*c + 3*d*x))/3 - (A*a*(3*c + 3*d*x + 10))/ 3 + 2*A*a*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^3) - A*a*x