Integrand size = 21, antiderivative size = 79 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=b^3 x-\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f} \]
b^3*x-1/2*a*(a^2+6*b^2)*arctanh(cos(f*x+e))/f-5/2*a^2*b*cot(f*x+e)/f-1/2*a ^2*cot(f*x+e)*csc(f*x+e)*(a+b*sin(f*x+e))/f
Time = 2.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.92 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {8 b^3 e+8 b^3 f x-12 a^2 b \cot \left (\frac {1}{2} (e+f x)\right )-a^3 \csc ^2\left (\frac {1}{2} (e+f x)\right )-4 a^3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-24 a b^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+4 a^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+24 a b^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+a^3 \sec ^2\left (\frac {1}{2} (e+f x)\right )+12 a^2 b \tan \left (\frac {1}{2} (e+f x)\right )}{8 f} \]
(8*b^3*e + 8*b^3*f*x - 12*a^2*b*Cot[(e + f*x)/2] - a^3*Csc[(e + f*x)/2]^2 - 4*a^3*Log[Cos[(e + f*x)/2]] - 24*a*b^2*Log[Cos[(e + f*x)/2]] + 4*a^3*Log [Sin[(e + f*x)/2]] + 24*a*b^2*Log[Sin[(e + f*x)/2]] + a^3*Sec[(e + f*x)/2] ^2 + 12*a^2*b*Tan[(e + f*x)/2])/(8*f)
Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3271, 3042, 3500, 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.
\(\displaystyle \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{\sin (e+f x)^3}dx\) |
\(\Big \downarrow \) 3271 |
\(\displaystyle \frac {1}{2} \int \csc ^2(e+f x) \left (2 \sin ^2(e+f x) b^3+5 a^2 b+a \left (a^2+6 b^2\right ) \sin (e+f x)\right )dx-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {2 \sin (e+f x)^2 b^3+5 a^2 b+a \left (a^2+6 b^2\right ) \sin (e+f x)}{\sin (e+f x)^2}dx-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \frac {1}{2} \left (\int \csc (e+f x) \left (2 \sin (e+f x) b^3+a \left (a^2+6 b^2\right )\right )dx-\frac {5 a^2 b \cot (e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\int \frac {2 \sin (e+f x) b^3+a \left (a^2+6 b^2\right )}{\sin (e+f x)}dx-\frac {5 a^2 b \cot (e+f x)}{f}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {1}{2} \left (a \left (a^2+6 b^2\right ) \int \csc (e+f x)dx-\frac {5 a^2 b \cot (e+f x)}{f}+2 b^3 x\right )-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (a \left (a^2+6 b^2\right ) \int \csc (e+f x)dx-\frac {5 a^2 b \cot (e+f x)}{f}+2 b^3 x\right )-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{2} \left (-\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (e+f x))}{f}-\frac {5 a^2 b \cot (e+f x)}{f}+2 b^3 x\right )-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}\) |
(2*b^3*x - (a*(a^2 + 6*b^2)*ArcTanh[Cos[e + f*x]])/f - (5*a^2*b*Cot[e + f* x])/f)/2 - (a^2*Cot[e + f*x]*Csc[e + f*x]*(a + b*Sin[e + f*x]))/(2*f)
3.2.72.3.1 Defintions of rubi rules used
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[((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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a^{2} b \cot \left (f x +e \right )+3 a \,b^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{3} \left (f x +e \right )}{f}\) | \(86\) |
default | \(\frac {a^{3} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a^{2} b \cot \left (f x +e \right )+3 a \,b^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{3} \left (f x +e \right )}{f}\) | \(86\) |
parallelrisch | \(\frac {4 \left (a^{3}+6 a \,b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+8 b^{3} f x -12 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b +12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b}{8 f}\) | \(97\) |
risch | \(b^{3} x -\frac {i a^{2} \left (i a \,{\mathrm e}^{3 i \left (f x +e \right )}+i a \,{\mathrm e}^{i \left (f x +e \right )}+6 b \,{\mathrm e}^{2 i \left (f x +e \right )}-6 b \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{f}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{f}\) | \(153\) |
norman | \(\frac {b^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+b^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {a^{3}}{8 f}+\frac {a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+3 b^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 b^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {3 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {7 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {11 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {3 a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f}-\frac {3 a^{2} b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} b \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {a \left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\) | \(283\) |
1/f*(a^3*(-1/2*csc(f*x+e)*cot(f*x+e)+1/2*ln(-cot(f*x+e)+csc(f*x+e)))-3*a^2 *b*cot(f*x+e)+3*a*b^2*ln(-cot(f*x+e)+csc(f*x+e))+b^3*(f*x+e))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (73) = 146\).
Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.96 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {4 \, b^{3} f x \cos \left (f x + e\right )^{2} - 4 \, b^{3} f x + 12 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, a^{3} \cos \left (f x + e\right ) + {\left (a^{3} + 6 \, a b^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (a^{3} + 6 \, a b^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{4 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \]
1/4*(4*b^3*f*x*cos(f*x + e)^2 - 4*b^3*f*x + 12*a^2*b*cos(f*x + e)*sin(f*x + e) + 2*a^3*cos(f*x + e) + (a^3 + 6*a*b^2 - (a^3 + 6*a*b^2)*cos(f*x + e)^ 2)*log(1/2*cos(f*x + e) + 1/2) - (a^3 + 6*a*b^2 - (a^3 + 6*a*b^2)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^2 - f)
\[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \csc ^{3}{\left (e + f x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {4 \, {\left (f x + e\right )} b^{3} + a^{3} {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 6 \, a b^{2} {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {12 \, a^{2} b}{\tan \left (f x + e\right )}}{4 \, f} \]
1/4*(4*(f*x + e)*b^3 + a^3*(2*cos(f*x + e)/(cos(f*x + e)^2 - 1) - log(cos( f*x + e) + 1) + log(cos(f*x + e) - 1)) - 6*a*b^2*(log(cos(f*x + e) + 1) - log(cos(f*x + e) - 1)) - 12*a^2*b/tan(f*x + e))/f
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.70 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, {\left (f x + e\right )} b^{3} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{8 \, f} \]
1/8*(a^3*tan(1/2*f*x + 1/2*e)^2 + 8*(f*x + e)*b^3 + 12*a^2*b*tan(1/2*f*x + 1/2*e) + 4*(a^3 + 6*a*b^2)*log(abs(tan(1/2*f*x + 1/2*e))) - (6*a^3*tan(1/ 2*f*x + 1/2*e)^2 + 36*a*b^2*tan(1/2*f*x + 1/2*e)^2 + 12*a^2*b*tan(1/2*f*x + 1/2*e) + a^3)/tan(1/2*f*x + 1/2*e)^2)/f
Time = 6.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.96 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a\,b^2+2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a\,b^2-2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}\right )}{f}-\frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}+\frac {a^3\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{2\,f}-\frac {3\,a^2\,b\,\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2\,f}+\frac {3\,a\,b^2\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2\,f} \]
(2*b^3*atan((2*b^3*cos(e/2 + (f*x)/2) + a^3*sin(e/2 + (f*x)/2) + 6*a*b^2*s in(e/2 + (f*x)/2))/(a^3*cos(e/2 + (f*x)/2) - 2*b^3*sin(e/2 + (f*x)/2) + 6* a*b^2*cos(e/2 + (f*x)/2))))/f - (a^3*cot(e/2 + (f*x)/2)^2)/(8*f) + (a^3*ta n(e/2 + (f*x)/2)^2)/(8*f) + (a^3*log(sin(e/2 + (f*x)/2)/cos(e/2 + (f*x)/2) ))/(2*f) - (3*a^2*b*cot(e/2 + (f*x)/2))/(2*f) + (3*a*b^2*log(sin(e/2 + (f* x)/2)/cos(e/2 + (f*x)/2)))/f + (3*a^2*b*tan(e/2 + (f*x)/2))/(2*f)