Integrand size = 27, antiderivative size = 187 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3}{2} b \left (2 a^2-b^2\right ) x-\frac {3 a \left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}-\frac {b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{8 a d}-\frac {13 b^3 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {17 b \cot (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d} \] Output:
3/2*b*(2*a^2-b^2)*x-3/8*a*(a^2-12*b^2)*arctanh(cos(d*x+c))/d-1/8*b^2*(73*a ^2-2*b^2)*cos(d*x+c)/a/d-13/4*b^3*cos(d*x+c)*sin(d*x+c)/d+17/8*b*cot(d*x+c )*(a+b*sin(d*x+c))^2/d+5/8*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^3/d-1/4* cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^4/a/d
Leaf count is larger than twice the leaf count of optimal. \(381\) vs. \(2(187)=374\).
Time = 7.81 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.04 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 b \left (-2 a^2+b^2\right ) (c+d x)}{2 d}-\frac {3 a b^2 \cos (c+d x)}{d}+\frac {\left (4 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )-b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{2 d}+\frac {\left (5 a^3-12 a b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a^2 b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 \left (a^3-12 a b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 \left (a^3-12 a b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\left (-5 a^3+12 a b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a^3 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-4 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )+b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {b^3 \sin (2 (c+d x))}{4 d}+\frac {a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{8 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]
Output:
(-3*b*(-2*a^2 + b^2)*(c + d*x))/(2*d) - (3*a*b^2*Cos[c + d*x])/d + ((4*a^2 *b*Cos[(c + d*x)/2] - b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*d) + ((5* a^3 - 12*a*b^2)*Csc[(c + d*x)/2]^2)/(32*d) - (a^2*b*Cot[(c + d*x)/2]*Csc[( c + d*x)/2]^2)/(8*d) - (a^3*Csc[(c + d*x)/2]^4)/(64*d) - (3*(a^3 - 12*a*b^ 2)*Log[Cos[(c + d*x)/2]])/(8*d) + (3*(a^3 - 12*a*b^2)*Log[Sin[(c + d*x)/2] ])/(8*d) + ((-5*a^3 + 12*a*b^2)*Sec[(c + d*x)/2]^2)/(32*d) + (a^3*Sec[(c + d*x)/2]^4)/(64*d) + (Sec[(c + d*x)/2]*(-4*a^2*b*Sin[(c + d*x)/2] + b^3*Si n[(c + d*x)/2]))/(2*d) - (b^3*Sin[2*(c + d*x)])/(4*d) + (a^2*b*Sec[(c + d* x)/2]^2*Tan[(c + d*x)/2])/(8*d)
Time = 1.59 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3372, 27, 3042, 3526, 3042, 3526, 25, 3042, 3512, 27, 3042, 3502, 27, 3042, 3214, 3042, 4257}
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.
| \(\displaystyle \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^3}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int 3 \csc ^3(c+d x) (a+b \sin (c+d x))^3 \left (-4 \sin ^2(c+d x) a^2+5 a^2+b \sin (c+d x) a\right )dx}{12 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^3 \left (-4 \sin ^2(c+d x) a^2+5 a^2+b \sin (c+d x) a\right )dx}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^3 \left (-4 \sin (c+d x)^2 a^2+5 a^2+b \sin (c+d x) a\right )}{\sin (c+d x)^3}dx}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc ^2(c+d x) (a+b \sin (c+d x))^2 \left (-18 b \sin ^2(c+d x) a^2+17 b a^2-\left (3 a^2-2 b^2\right ) \sin (c+d x) a\right )dx-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {(a+b \sin (c+d x))^2 \left (-18 b \sin (c+d x)^2 a^2+17 b a^2-\left (3 a^2-2 b^2\right ) \sin (c+d x) a\right )}{\sin (c+d x)^2}dx-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\int -\csc (c+d x) (a+b \sin (c+d x)) \left (52 b^2 \sin ^2(c+d x) a^2+3 \left (a^2-12 b^2\right ) a^2+b \left (21 a^2-2 b^2\right ) \sin (c+d x) a\right )dx-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\int \csc (c+d x) (a+b \sin (c+d x)) \left (52 b^2 \sin ^2(c+d x) a^2+3 \left (a^2-12 b^2\right ) a^2+b \left (21 a^2-2 b^2\right ) \sin (c+d x) a\right )dx-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\int \frac {(a+b \sin (c+d x)) \left (52 b^2 \sin (c+d x)^2 a^2+3 \left (a^2-12 b^2\right ) a^2+b \left (21 a^2-2 b^2\right ) \sin (c+d x) a\right )}{\sin (c+d x)}dx-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{2} \int 2 \csc (c+d x) \left (3 \left (a^2-12 b^2\right ) a^3+12 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2+b^2 \left (73 a^2-2 b^2\right ) \sin ^2(c+d x) a\right )dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\int \csc (c+d x) \left (3 \left (a^2-12 b^2\right ) a^3+12 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2+b^2 \left (73 a^2-2 b^2\right ) \sin ^2(c+d x) a\right )dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\int \frac {3 \left (a^2-12 b^2\right ) a^3+12 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2+b^2 \left (73 a^2-2 b^2\right ) \sin (c+d x)^2 a}{\sin (c+d x)}dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\int 3 \csc (c+d x) \left (\left (a^2-12 b^2\right ) a^3+4 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2\right )dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-3 \int \csc (c+d x) \left (\left (a^2-12 b^2\right ) a^3+4 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2\right )dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-3 \int \frac {\left (a^2-12 b^2\right ) a^3+4 b \left (2 a^2-b^2\right ) \sin (c+d x) a^2}{\sin (c+d x)}dx+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-3 \left (a^3 \left (a^2-12 b^2\right ) \int \csc (c+d x)dx+4 a^2 b x \left (2 a^2-b^2\right )\right )+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-3 \left (a^3 \left (a^2-12 b^2\right ) \int \csc (c+d x)dx+4 a^2 b x \left (2 a^2-b^2\right )\right )+\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {26 a^2 b^3 \sin (c+d x) \cos (c+d x)}{d}+\frac {a b^2 \left (73 a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {17 a^2 b \cot (c+d x) (a+b \sin (c+d x))^2}{d}-3 \left (4 a^2 b x \left (2 a^2-b^2\right )-\frac {a^3 \left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{d}\right )\right )-\frac {5 a^2 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}}{4 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]
Output:
-1/4*(Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(a*d) - ((-5*a^2 *Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*d) + (-3*(4*a^2*b*(2 *a^2 - b^2)*x - (a^3*(a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/d) + (a*b^2*(73 *a^2 - 2*b^2)*Cos[c + d*x])/d + (26*a^2*b^3*Cos[c + d*x]*Sin[c + d*x])/d - (17*a^2*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^2)/d)/2)/(4*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(226\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(226\) |
| risch | \(3 a^{2} b x -\frac {3 b^{3} x}{2}+\frac {i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{2 d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{2}}{2 d}-\frac {i b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i \left (-3 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-48 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+8 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-5 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-5 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+96 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-24 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-80 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+32 a^{2} b -8 b^{3}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d}\) | \(411\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/4/sin(d*x+c)^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c)^5+1/8 *cos(d*x+c)^3+3/8*cos(d*x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c)))+3*a^2*b*(-1/3* cot(d*x+c)^3+cot(d*x+c)+d*x+c)+3*a*b^2*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2 *cos(d*x+c)^3-3/2*cos(d*x+c)-3/2*ln(csc(d*x+c)-cot(d*x+c)))+b^3*(-1/sin(d* x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x-3/2*c))
Time = 0.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.77 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {48 \, a b^{2} \cos \left (d x + c\right )^{5} - 24 \, {\left (2 \, a^{2} b - b^{3}\right )} d x \cos \left (d x + c\right )^{4} + 48 \, {\left (2 \, a^{2} b - b^{3}\right )} d x \cos \left (d x + c\right )^{2} + 10 \, {\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left (2 \, a^{2} b - b^{3}\right )} d x - 6 \, {\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} - 12 \, a b^{2} - 2 \, {\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} - 12 \, a b^{2} - 2 \, {\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (b^{3} \cos \left (d x + c\right )^{5} + 4 \, {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/16*(48*a*b^2*cos(d*x + c)^5 - 24*(2*a^2*b - b^3)*d*x*cos(d*x + c)^4 + 4 8*(2*a^2*b - b^3)*d*x*cos(d*x + c)^2 + 10*(a^3 - 12*a*b^2)*cos(d*x + c)^3 - 24*(2*a^2*b - b^3)*d*x - 6*(a^3 - 12*a*b^2)*cos(d*x + c) + 3*((a^3 - 12* a*b^2)*cos(d*x + c)^4 + a^3 - 12*a*b^2 - 2*(a^3 - 12*a*b^2)*cos(d*x + c)^2 )*log(1/2*cos(d*x + c) + 1/2) - 3*((a^3 - 12*a*b^2)*cos(d*x + c)^4 + a^3 - 12*a*b^2 - 2*(a^3 - 12*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2 ) + 8*(b^3*cos(d*x + c)^5 + 4*(2*a^2*b - b^3)*cos(d*x + c)^3 - 3*(2*a^2*b - b^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.13 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} b - 8 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} b^{3} - a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, a b^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/16*(16*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^2*b - 8*( 3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*b^3 - a^3*(2*(5*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) + 12*a*b^2 *(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c ) + 1) - 3*log(cos(d*x + c) - 1)))/d
Time = 0.23 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.83 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, {\left (2 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} + 24 \, {\left (a^{3} - 12 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {64 \, {\left (b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(a^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 8*a^3* tan(1/2*d*x + 1/2*c)^2 + 24*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 120*a^2*b*tan(1 /2*d*x + 1/2*c) + 32*b^3*tan(1/2*d*x + 1/2*c) + 96*(2*a^2*b - b^3)*(d*x + c) + 24*(a^3 - 12*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) + 64*(b^3*tan(1/2* d*x + 1/2*c)^3 - 6*a*b^2*tan(1/2*d*x + 1/2*c)^2 - b^3*tan(1/2*d*x + 1/2*c) - 6*a*b^2)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - (50*a^3*tan(1/2*d*x + 1/2*c)^ 4 - 600*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 120*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 32*b^3*tan(1/2*d*x + 1/2*c)^3 - 8*a^3*tan(1/2*d*x + 1/2*c)^2 + 24*a*b^2*ta n(1/2*d*x + 1/2*c)^2 + 8*a^2*b*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1 /2*c)^4)/d
Time = 34.27 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.74 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int((cot(c + d*x)^4*(a + b*sin(c + d*x))^3)/sin(c + d*x),x)
Output:
(a^3*tan(c/2 + (d*x)/2)^4)/(64*d) + (tan(c/2 + (d*x)/2)^2*((3*a*b^2)/8 - a ^3/8))/d - (tan(c/2 + (d*x)/2)^2*(6*a*b^2 - (3*a^3)/2) + tan(c/2 + (d*x)/2 )^6*(102*a*b^2 - 2*a^3) + tan(c/2 + (d*x)/2)^4*(108*a*b^2 - (15*a^3)/4) - tan(c/2 + (d*x)/2)^3*(26*a^2*b - 8*b^3) - tan(c/2 + (d*x)/2)^7*(30*a^2*b + 8*b^3) - tan(c/2 + (d*x)/2)^5*(58*a^2*b - 32*b^3) + a^3/4 + 2*a^2*b*tan(c /2 + (d*x)/2))/(d*(16*tan(c/2 + (d*x)/2)^4 + 32*tan(c/2 + (d*x)/2)^6 + 16* tan(c/2 + (d*x)/2)^8)) - (tan(c/2 + (d*x)/2)*((15*a^2*b)/8 - b^3/2))/d + ( 3*a*log(tan(c/2 + (d*x)/2))*(a^2 - 12*b^2))/(8*d) + (a^2*b*tan(c/2 + (d*x) /2)^3)/(8*d) - (3*b*atan(((3*b*(2*a^2 - b^2)*(tan(c/2 + (d*x)/2)*(9*a*b^2 - (3*a^3)/4) - 6*a^2*b + 3*b^3 - b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2)*9i))/2 + (3*b*(2*a^2 - b^2)*(tan(c/2 + (d*x)/2)*(9*a*b^2 - (3*a^3)/4) - 6*a^2*b + 3*b^3 + b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2)*9i))/2)/(2*tan(c/2 + (d*x)/2) *(9*b^6 - 36*a^2*b^4 + 36*a^4*b^2) + 27*a*b^5 + (9*a^5*b)/2 - (225*a^3*b^3 )/4 - (b*(2*a^2 - b^2)*(tan(c/2 + (d*x)/2)*(9*a*b^2 - (3*a^3)/4) - 6*a^2*b + 3*b^3 - b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2)*9i)*3i)/2 + (b*(2*a^2 - b^2) *(tan(c/2 + (d*x)/2)*(9*a*b^2 - (3*a^3)/4) - 6*a^2*b + 3*b^3 + b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2)*9i)*3i)/2))*(2*a^2 - b^2))/d
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.42 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}-192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-96 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-64 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -16 \cos \left (d x +c \right ) a^{3}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{3}-288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a \,b^{2}-15 \sin \left (d x +c \right )^{4} a^{3}+192 \sin \left (d x +c \right )^{4} a^{2} b d x +240 \sin \left (d x +c \right )^{4} a \,b^{2}-96 \sin \left (d x +c \right )^{4} b^{3} d x}{64 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x)
Output:
( - 32*cos(c + d*x)*sin(c + d*x)**5*b**3 - 192*cos(c + d*x)*sin(c + d*x)** 4*a*b**2 + 256*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 64*cos(c + d*x)*sin(c + d*x)**3*b**3 + 40*cos(c + d*x)*sin(c + d*x)**2*a**3 - 96*cos(c + d*x)*s in(c + d*x)**2*a*b**2 - 64*cos(c + d*x)*sin(c + d*x)*a**2*b - 16*cos(c + d *x)*a**3 + 24*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**3 - 288*log(tan((c + d*x)/2))*sin(c + d*x)**4*a*b**2 - 15*sin(c + d*x)**4*a**3 + 192*sin(c + d*x)**4*a**2*b*d*x + 240*sin(c + d*x)**4*a*b**2 - 96*sin(c + d*x)**4*b**3* d*x)/(64*sin(c + d*x)**4*d)