Integrand size = 21, antiderivative size = 109 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \]
-1/2*b*(3*a^2+2*b^2)*arctanh(cos(f*x+e))/f-1/3*a*(2*a^2+9*b^2)*cot(f*x+e)/ f-7/6*a^2*b*cot(f*x+e)*csc(f*x+e)/f-1/3*a^2*cot(f*x+e)*csc(f*x+e)^2*(a+b*s in(f*x+e))/f
Leaf count is larger than twice the leaf count of optimal. \(525\) vs. \(2(109)=218\).
Time = 7.68 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.82 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\left (-2 a^3 \cos \left (\frac {1}{2} (e+f x)\right )-9 a b^2 \cos \left (\frac {1}{2} (e+f x)\right )\right ) \csc \left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{6 f (a+b \sin (e+f x))^3}-\frac {3 a^2 b \csc ^2\left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{8 f (a+b \sin (e+f x))^3}-\frac {a^3 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{24 f (a+b \sin (e+f x))^3}+\frac {\left (-3 a^2 b-2 b^3\right ) (b+a \csc (e+f x))^3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{2 f (a+b \sin (e+f x))^3}+\frac {\left (3 a^2 b+2 b^3\right ) (b+a \csc (e+f x))^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{2 f (a+b \sin (e+f x))^3}+\frac {3 a^2 b (b+a \csc (e+f x))^3 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x)}{8 f (a+b \sin (e+f x))^3}+\frac {(b+a \csc (e+f x))^3 \sec \left (\frac {1}{2} (e+f x)\right ) \left (2 a^3 \sin \left (\frac {1}{2} (e+f x)\right )+9 a b^2 \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{6 f (a+b \sin (e+f x))^3}+\frac {a^3 (b+a \csc (e+f x))^3 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x) \tan \left (\frac {1}{2} (e+f x)\right )}{24 f (a+b \sin (e+f x))^3} \]
((-2*a^3*Cos[(e + f*x)/2] - 9*a*b^2*Cos[(e + f*x)/2])*Csc[(e + f*x)/2]*(b + a*Csc[e + f*x])^3*Sin[e + f*x]^3)/(6*f*(a + b*Sin[e + f*x])^3) - (3*a^2* b*Csc[(e + f*x)/2]^2*(b + a*Csc[e + f*x])^3*Sin[e + f*x]^3)/(8*f*(a + b*Si n[e + f*x])^3) - (a^3*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*(b + a*Csc[e + f *x])^3*Sin[e + f*x]^3)/(24*f*(a + b*Sin[e + f*x])^3) + ((-3*a^2*b - 2*b^3) *(b + a*Csc[e + f*x])^3*Log[Cos[(e + f*x)/2]]*Sin[e + f*x]^3)/(2*f*(a + b* Sin[e + f*x])^3) + ((3*a^2*b + 2*b^3)*(b + a*Csc[e + f*x])^3*Log[Sin[(e + f*x)/2]]*Sin[e + f*x]^3)/(2*f*(a + b*Sin[e + f*x])^3) + (3*a^2*b*(b + a*Cs c[e + f*x])^3*Sec[(e + f*x)/2]^2*Sin[e + f*x]^3)/(8*f*(a + b*Sin[e + f*x]) ^3) + ((b + a*Csc[e + f*x])^3*Sec[(e + f*x)/2]*(2*a^3*Sin[(e + f*x)/2] + 9 *a*b^2*Sin[(e + f*x)/2])*Sin[e + f*x]^3)/(6*f*(a + b*Sin[e + f*x])^3) + (a ^3*(b + a*Csc[e + f*x])^3*Sec[(e + f*x)/2]^2*Sin[e + f*x]^3*Tan[(e + f*x)/ 2])/(24*f*(a + b*Sin[e + f*x])^3)
Time = 0.67 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3271, 3042, 3500, 3042, 3227, 3042, 4254, 24, 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.
| \(\displaystyle \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{\sin (e+f x)^4}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{3} \int \csc ^3(e+f x) \left (7 b a^2+\left (2 a^2+9 b^2\right ) \sin (e+f x) a+b \left (a^2+3 b^2\right ) \sin ^2(e+f x)\right )dx-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {7 b a^2+\left (2 a^2+9 b^2\right ) \sin (e+f x) a+b \left (a^2+3 b^2\right ) \sin (e+f x)^2}{\sin (e+f x)^3}dx-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \csc ^2(e+f x) \left (2 a \left (2 a^2+9 b^2\right )+3 b \left (3 a^2+2 b^2\right ) \sin (e+f x)\right )dx-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {2 a \left (2 a^2+9 b^2\right )+3 b \left (3 a^2+2 b^2\right ) \sin (e+f x)}{\sin (e+f x)^2}dx-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (2 a \left (2 a^2+9 b^2\right ) \int \csc ^2(e+f x)dx+3 b \left (3 a^2+2 b^2\right ) \int \csc (e+f x)dx\right )-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 b \left (3 a^2+2 b^2\right ) \int \csc (e+f x)dx+2 a \left (2 a^2+9 b^2\right ) \int \csc (e+f x)^2dx\right )-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 b \left (3 a^2+2 b^2\right ) \int \csc (e+f x)dx-\frac {2 a \left (2 a^2+9 b^2\right ) \int 1d\cot (e+f x)}{f}\right )-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 b \left (3 a^2+2 b^2\right ) \int \csc (e+f x)dx-\frac {2 a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{f}\right )-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {3 b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{f}-\frac {2 a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{f}\right )-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\) |
(((-3*b*(3*a^2 + 2*b^2)*ArcTanh[Cos[e + f*x]])/f - (2*a*(2*a^2 + 9*b^2)*Co t[e + f*x])/f)/2 - (7*a^2*b*Cot[e + f*x]*Csc[e + f*x])/(2*f))/3 - (a^2*Cot [e + f*x]*Csc[e + f*x]^2*(a + b*Sin[e + f*x]))/(3*f)
3.2.73.3.1 Defintions of rubi rules used
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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.85 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a^{2} b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a \,b^{2} \cot \left (f x +e \right )+b^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{f}\) | \(99\) |
| default | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a^{2} b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a \,b^{2} \cot \left (f x +e \right )+b^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{f}\) | \(99\) |
| parallelrisch | \(\frac {\left (36 a^{2} b +24 b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+a \left (a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+9 b \right ) \cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+9 a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+9 a^{2}+36 b^{2}\right ) a}{24 f}\) | \(134\) |
| risch | \(\frac {a \left (-18 i b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+9 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+12 i a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+36 i b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 i a^{2}-18 i b^{2}-9 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}\) | \(186\) |
| norman | \(\frac {-\frac {a^{3}}{24 f}+\frac {a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {21 a^{2} b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {9 a^{2} b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {33 a^{2} b \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a \left (a^{2}+3 b^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a \left (a^{2}+3 b^{2}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {a \left (7 a^{2}+24 b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a \left (7 a^{2}+24 b^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {3 a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {3 a^{2} b \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {b \left (3 a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\) | \(290\) |
1/f*(a^3*(-2/3-1/3*csc(f*x+e)^2)*cot(f*x+e)+3*a^2*b*(-1/2*csc(f*x+e)*cot(f *x+e)+1/2*ln(-cot(f*x+e)+csc(f*x+e)))-3*a*b^2*cot(f*x+e)+b^3*ln(-cot(f*x+e )+csc(f*x+e)))
Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.75 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {18 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 4 \, {\left (2 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) + 12 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right )}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
1/12*(18*a^2*b*cos(f*x + e)*sin(f*x + e) - 4*(2*a^3 + 9*a*b^2)*cos(f*x + e )^3 + 3*(3*a^2*b + 2*b^3 - (3*a^2*b + 2*b^3)*cos(f*x + e)^2)*log(1/2*cos(f *x + e) + 1/2)*sin(f*x + e) - 3*(3*a^2*b + 2*b^3 - (3*a^2*b + 2*b^3)*cos(f *x + e)^2)*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) + 12*(a^3 + 3*a*b^2)* cos(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))
\[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \csc ^{4}{\left (e + f x \right )}\, dx \]
Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.08 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {9 \, a^{2} b {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 6 \, b^{3} {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {36 \, a b^{2}}{\tan \left (f x + e\right )} - \frac {4 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{3}}{\tan \left (f x + e\right )^{3}}}{12 \, f} \]
1/12*(9*a^2*b*(2*cos(f*x + e)/(cos(f*x + e)^2 - 1) - log(cos(f*x + e) + 1) + log(cos(f*x + e) - 1)) - 6*b^3*(log(cos(f*x + e) + 1) - log(cos(f*x + e ) - 1)) - 36*a*b^2/tan(f*x + e) - 4*(3*tan(f*x + e)^2 + 1)*a^3/tan(f*x + e )^3)/f
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.74 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {66 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 44 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
1/24*(a^3*tan(1/2*f*x + 1/2*e)^3 + 9*a^2*b*tan(1/2*f*x + 1/2*e)^2 + 9*a^3* tan(1/2*f*x + 1/2*e) + 36*a*b^2*tan(1/2*f*x + 1/2*e) + 12*(3*a^2*b + 2*b^3 )*log(abs(tan(1/2*f*x + 1/2*e))) - (66*a^2*b*tan(1/2*f*x + 1/2*e)^3 + 44*b ^3*tan(1/2*f*x + 1/2*e)^3 + 9*a^3*tan(1/2*f*x + 1/2*e)^2 + 36*a*b^2*tan(1/ 2*f*x + 1/2*e)^2 + 9*a^2*b*tan(1/2*f*x + 1/2*e) + a^3)/tan(1/2*f*x + 1/2*e )^3)/f
Time = 6.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.38 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{2}+b^3\right )}{f}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,a^3+12\,a\,b^2\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{f}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f} \]