Integrand size = 21, antiderivative size = 134 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {3 a \left (a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a \left (a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f} \] Output:
-3/8*a*(a^2+4*b^2)*arctanh(cos(f*x+e))/f-b*(2*a^2+b^2)*cot(f*x+e)/f-3/8*a* (a^2+4*b^2)*cot(f*x+e)*csc(f*x+e)/f-3/4*a^2*b*cot(f*x+e)*csc(f*x+e)^2/f-1/ 4*a^2*cot(f*x+e)*csc(f*x+e)^3*(a+b*sin(f*x+e))/f
Leaf count is larger than twice the leaf count of optimal. \(322\) vs. \(2(134)=268\).
Time = 9.05 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.40 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\left (-2 a^2 b \cos \left (\frac {1}{2} (e+f x)\right )-b^3 \cos \left (\frac {1}{2} (e+f x)\right )\right ) \csc \left (\frac {1}{2} (e+f x)\right )}{2 f}-\frac {3 \left (a^3+4 a b^2\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {a^2 b \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {a^3 \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {3 \left (a^3+4 a b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {3 \left (a^3+4 a b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {3 \left (a^3+4 a b^2\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {a^3 \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (2 a^2 b \sin \left (\frac {1}{2} (e+f x)\right )+b^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {a^2 b \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{8 f} \] Input:
Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]
Output:
((-2*a^2*b*Cos[(e + f*x)/2] - b^3*Cos[(e + f*x)/2])*Csc[(e + f*x)/2])/(2*f ) - (3*(a^3 + 4*a*b^2)*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*b*Cot[(e + f*x)/2 ]*Csc[(e + f*x)/2]^2)/(8*f) - (a^3*Csc[(e + f*x)/2]^4)/(64*f) - (3*(a^3 + 4*a*b^2)*Log[Cos[(e + f*x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)*Log[Sin[(e + f* x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)*Sec[(e + f*x)/2]^2)/(32*f) + (a^3*Sec[( e + f*x)/2]^4)/(64*f) + (Sec[(e + f*x)/2]*(2*a^2*b*Sin[(e + f*x)/2] + b^3* Sin[(e + f*x)/2]))/(2*f) + (a^2*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(8* f)
Time = 0.82 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3271, 3042, 3500, 27, 3042, 3227, 3042, 4254, 24, 4255, 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.
| \(\displaystyle \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{\sin (e+f x)^5}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{4} \int \csc ^4(e+f x) \left (9 b a^2+3 \left (a^2+4 b^2\right ) \sin (e+f x) a+2 b \left (a^2+2 b^2\right ) \sin ^2(e+f x)\right )dx-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {9 b a^2+3 \left (a^2+4 b^2\right ) \sin (e+f x) a+2 b \left (a^2+2 b^2\right ) \sin (e+f x)^2}{\sin (e+f x)^4}dx-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int 3 \csc ^3(e+f x) \left (3 a \left (a^2+4 b^2\right )+4 b \left (2 a^2+b^2\right ) \sin (e+f x)\right )dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\int \csc ^3(e+f x) \left (3 a \left (a^2+4 b^2\right )+4 b \left (2 a^2+b^2\right ) \sin (e+f x)\right )dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {3 a \left (a^2+4 b^2\right )+4 b \left (2 a^2+b^2\right ) \sin (e+f x)}{\sin (e+f x)^3}dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{4} \left (3 a \left (a^2+4 b^2\right ) \int \csc ^3(e+f x)dx+4 b \left (2 a^2+b^2\right ) \int \csc ^2(e+f x)dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (4 b \left (2 a^2+b^2\right ) \int \csc (e+f x)^2dx+3 a \left (a^2+4 b^2\right ) \int \csc (e+f x)^3dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4 b \left (2 a^2+b^2\right ) \int 1d\cot (e+f x)}{f}+3 a \left (a^2+4 b^2\right ) \int \csc (e+f x)^3dx-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (3 a \left (a^2+4 b^2\right ) \int \csc (e+f x)^3dx-\frac {4 b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{4} \left (3 a \left (a^2+4 b^2\right ) \left (\frac {1}{2} \int \csc (e+f x)dx-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {4 b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (3 a \left (a^2+4 b^2\right ) \left (\frac {1}{2} \int \csc (e+f x)dx-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {4 b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (3 a \left (a^2+4 b^2\right ) \left (-\frac {\text {arctanh}(\cos (e+f x))}{2 f}-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {4 b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}\) |
Input:
Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]
Output:
((-4*b*(2*a^2 + b^2)*Cot[e + f*x])/f - (3*a^2*b*Cot[e + f*x]*Csc[e + f*x]^ 2)/f + 3*a*(a^2 + 4*b^2)*(-1/2*ArcTanh[Cos[e + f*x]]/f - (Cot[e + f*x]*Csc [e + f*x])/(2*f)))/4 - (a^2*Cot[e + f*x]*Csc[e + f*x]^3*(a + b*Sin[e + f*x ]))/(4*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((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 - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )-\cot \left (f x +e \right ) b^{3}}{f}\) | \(129\) |
| default | \(\frac {a^{3} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )-\cot \left (f x +e \right ) b^{3}}{f}\) | \(129\) |
| parallelrisch | \(\frac {-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} a^{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} a^{3}-8 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} a^{2} b +8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} a^{2} b -8 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{3}-24 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a \,b^{2}+8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{3}+24 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a \,b^{2}-72 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b -32 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{3}+72 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b +32 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{3}+24 a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+96 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a \,b^{2}}{64 f}\) | \(227\) |
| risch | \(-\frac {i \left (3 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}+12 i a \,b^{2} {\mathrm e}^{7 i \left (f x +e \right )}-11 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-12 i a \,b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+8 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-11 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-12 i a \,b^{2} {\mathrm e}^{3 i \left (f x +e \right )}-48 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-24 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+3 i a^{3} {\mathrm e}^{i \left (f x +e \right )}+12 i a \,b^{2} {\mathrm e}^{i \left (f x +e \right )}+64 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+24 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}-16 a^{2} b -8 b^{3}\right )}{4 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{2 f}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{2 f}\) | \(311\) |
| norman | \(\frac {-\frac {a^{3}}{64 f}+\frac {a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{64 f}-\frac {\left (8 a^{3}+21 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{8 f}-\frac {\left (27 a^{3}+72 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{32 f}-\frac {\left (49 a^{3}+132 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{32 f}-\frac {a \left (11 a^{2}+24 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{64 f}+\frac {a \left (11 a^{2}+24 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{64 f}-\frac {a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{8 f}-\frac {b \left (3 a^{2}+b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 f}+\frac {b \left (3 a^{2}+b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{2 f}-\frac {b \left (21 a^{2}+8 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{8 f}+\frac {b \left (21 a^{2}+8 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {3 a \left (a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}\) | \(368\) |
Input:
int(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
1/f*(a^3*((-1/4*csc(f*x+e)^3-3/8*csc(f*x+e))*cot(f*x+e)+3/8*ln(csc(f*x+e)- cot(f*x+e)))+3*a^2*b*(-2/3-1/3*csc(f*x+e)^2)*cot(f*x+e)+3*a*b^2*(-1/2*csc( f*x+e)*cot(f*x+e)+1/2*ln(csc(f*x+e)-cot(f*x+e)))-cot(f*x+e)*b^3)
Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.78 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {6 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, a^{3} + 12 \, a b^{2}\right )} \cos \left (f x + e\right ) - 3 \, {\left ({\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 16 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{16 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \] Input:
integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="fricas")
Output:
1/16*(6*(a^3 + 4*a*b^2)*cos(f*x + e)^3 - 2*(5*a^3 + 12*a*b^2)*cos(f*x + e) - 3*((a^3 + 4*a*b^2)*cos(f*x + e)^4 + a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*c os(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 + 4*a*b^2)*cos(f*x + e)^4 + a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2) + 16*((2*a^2*b + b^3)*cos(f*x + e)^3 - (3*a^2*b + b^3)*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)
Timed out. \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)**5*(a+b*sin(f*x+e))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 12 \, a b^{2} {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {16 \, b^{3}}{\tan \left (f x + e\right )} - \frac {16 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} b}{\tan \left (f x + e\right )^{3}}}{16 \, f} \] Input:
integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="maxima")
Output:
1/16*(a^3*(2*(3*cos(f*x + e)^3 - 5*cos(f*x + e))/(cos(f*x + e)^4 - 2*cos(f *x + e)^2 + 1) - 3*log(cos(f*x + e) + 1) + 3*log(cos(f*x + e) - 1)) + 12*a *b^2*(2*cos(f*x + e)/(cos(f*x + e)^2 - 1) - log(cos(f*x + e) + 1) + log(co s(f*x + e) - 1)) - 16*b^3/tan(f*x + e) - 16*(3*tan(f*x + e)^2 + 1)*a^2*b/t an(f*x + e)^3)/f
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (126) = 252\).
Time = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.90 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, {\left (a^{3} + 4 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 200 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{64 \, f} \] Input:
integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="giac")
Output:
1/64*(a^3*tan(1/2*f*x + 1/2*e)^4 + 8*a^2*b*tan(1/2*f*x + 1/2*e)^3 + 8*a^3* tan(1/2*f*x + 1/2*e)^2 + 24*a*b^2*tan(1/2*f*x + 1/2*e)^2 + 72*a^2*b*tan(1/ 2*f*x + 1/2*e) + 32*b^3*tan(1/2*f*x + 1/2*e) + 24*(a^3 + 4*a*b^2)*log(abs( tan(1/2*f*x + 1/2*e))) - (50*a^3*tan(1/2*f*x + 1/2*e)^4 + 200*a*b^2*tan(1/ 2*f*x + 1/2*e)^4 + 72*a^2*b*tan(1/2*f*x + 1/2*e)^3 + 32*b^3*tan(1/2*f*x + 1/2*e)^3 + 8*a^3*tan(1/2*f*x + 1/2*e)^2 + 24*a*b^2*tan(1/2*f*x + 1/2*e)^2 + 8*a^2*b*tan(1/2*f*x + 1/2*e) + a^3)/tan(1/2*f*x + 1/2*e)^4)/f
Time = 17.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.51 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (18\,a^2\,b+8\,b^3\right )+\frac {a^3}{4}+2\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{16\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^3}{8}+\frac {3\,a\,b^2}{8}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{8}+\frac {b^3}{2}\right )}{f}+\frac {a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{8\,f} \] Input:
int((a + b*sin(e + f*x))^3/sin(e + f*x)^5,x)
Output:
(a^3*tan(e/2 + (f*x)/2)^4)/(64*f) - (cot(e/2 + (f*x)/2)^4*(tan(e/2 + (f*x) /2)^2*(6*a*b^2 + 2*a^3) + tan(e/2 + (f*x)/2)^3*(18*a^2*b + 8*b^3) + a^3/4 + 2*a^2*b*tan(e/2 + (f*x)/2)))/(16*f) + (tan(e/2 + (f*x)/2)^2*((3*a*b^2)/8 + a^3/8))/f + (log(tan(e/2 + (f*x)/2))*((3*a*b^2)/2 + (3*a^3)/8))/f + (ta n(e/2 + (f*x)/2)*((9*a^2*b)/8 + b^3/2))/f + (a^2*b*tan(e/2 + (f*x)/2)^3)/( 8*f)
Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.25 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {-16 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a^{2} b -8 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b^{3}-3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a^{3}-12 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a \,b^{2}-8 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a^{2} b -2 \cos \left (f x +e \right ) a^{3}+3 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{4} a^{3}+12 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{4} a \,b^{2}}{8 \sin \left (f x +e \right )^{4} f} \] Input:
int(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x)
Output:
( - 16*cos(e + f*x)*sin(e + f*x)**3*a**2*b - 8*cos(e + f*x)*sin(e + f*x)** 3*b**3 - 3*cos(e + f*x)*sin(e + f*x)**2*a**3 - 12*cos(e + f*x)*sin(e + f*x )**2*a*b**2 - 8*cos(e + f*x)*sin(e + f*x)*a**2*b - 2*cos(e + f*x)*a**3 + 3 *log(tan((e + f*x)/2))*sin(e + f*x)**4*a**3 + 12*log(tan((e + f*x)/2))*sin (e + f*x)**4*a*b**2)/(8*sin(e + f*x)**4*f)