Integrand size = 13, antiderivative size = 42 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=-\frac {3 \text {arctanh}(\cos (x))}{2 a}+\frac {2 \cot (x)}{a}-\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc (x)}{a+a \sin (x)} \] Output:
-3/2*arctanh(cos(x))/a+2*cot(x)/a-3/2*cot(x)*csc(x)/a+cot(x)*csc(x)/(a+a*s in(x))
Time = 0.48 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.98 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {4 \cot \left (\frac {x}{2}\right )-\csc ^2\left (\frac {x}{2}\right )-12 \log \left (\cos \left (\frac {x}{2}\right )\right )+12 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )-\frac {16 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}-4 \tan \left (\frac {x}{2}\right )}{8 a} \] Input:
Integrate[Csc[x]^3/(a + a*Sin[x]),x]
Output:
(4*Cot[x/2] - Csc[x/2]^2 - 12*Log[Cos[x/2]] + 12*Log[Sin[x/2]] + Sec[x/2]^ 2 - (16*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - 4*Tan[x/2])/(8*a)
Time = 0.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 3247, 25, 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.
\(\displaystyle \int \frac {\csc ^3(x)}{a \sin (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (x)^3 (a \sin (x)+a)}dx\) |
\(\Big \downarrow \) 3247 |
\(\displaystyle \frac {\cot (x) \csc (x)}{a \sin (x)+a}-\frac {\int -\csc ^3(x) (3 a-2 a \sin (x))dx}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \csc ^3(x) (3 a-2 a \sin (x))dx}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {3 a-2 a \sin (x)}{\sin (x)^3}dx}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {3 a \int \csc ^3(x)dx-2 a \int \csc ^2(x)dx}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 a \int \csc (x)^3dx-2 a \int \csc (x)^2dx}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {2 a \int 1d\cot (x)+3 a \int \csc (x)^3dx}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3 a \int \csc (x)^3dx+2 a \cot (x)}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3 a \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+2 a \cot (x)}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 a \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+2 a \cot (x)}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {3 a \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )+2 a \cot (x)}{a^2}+\frac {\cot (x) \csc (x)}{a \sin (x)+a}\) |
Input:
Int[Csc[x]^3/(a + a*Sin[x]),x]
Output:
(2*a*Cot[x] + 3*a*(-1/2*ArcTanh[Cos[x]] - (Cot[x]*Csc[x])/2))/a^2 + (Cot[x ]*Csc[x])/(a + a*Sin[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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 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 = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {x}{2}\right )}+6 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}-2 \tan \left (\frac {x}{2}\right )+\frac {8}{\tan \left (\frac {x}{2}\right )+1}}{4 a}\) | \(54\) |
parallelrisch | \(\frac {24+12 \ln \left (\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )+\tan \left (\frac {x}{2}\right )^{3}-\cot \left (\frac {x}{2}\right )^{2}-3 \tan \left (\frac {x}{2}\right )^{2}+3 \cot \left (\frac {x}{2}\right )}{8 a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(57\) |
norman | \(\frac {\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{a}-\frac {1}{8 a}+\frac {3 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {3 \tan \left (\frac {x}{2}\right )^{4}}{8 a}+\frac {\tan \left (\frac {x}{2}\right )^{5}}{8 a}}{\tan \left (\frac {x}{2}\right )^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a}\) | \(75\) |
risch | \(\frac {3 \,{\mathrm e}^{4 i x}-5 \,{\mathrm e}^{2 i x}+3 i {\mathrm e}^{3 i x}+4-i {\mathrm e}^{i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right ) a}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a}\) | \(83\) |
Input:
int(csc(x)^3/(a+a*sin(x)),x,method=_RETURNVERBOSE)
Output:
1/4/a*(-1/2/tan(1/2*x)^2+2/tan(1/2*x)+6*ln(tan(1/2*x))+1/2*tan(1/2*x)^2-2* tan(1/2*x)+8/(tan(1/2*x)+1))
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (38) = 76\).
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.19 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {8 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )^{2} - 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (4 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - 6 \, \cos \left (x\right ) - 4}{4 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) + {\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right ) - a\right )}} \] Input:
integrate(csc(x)^3/(a+a*sin(x)),x, algorithm="fricas")
Output:
1/4*(8*cos(x)^3 + 6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin (x) - cos(x) - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + cos(x)^2 + (cos(x) ^2 - 1)*sin(x) - cos(x) - 1)*log(-1/2*cos(x) + 1/2) - 2*(4*cos(x)^2 + cos( x) - 2)*sin(x) - 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) + (a*co s(x)^2 - a)*sin(x) - a)
\[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(csc(x)**3/(a+a*sin(x)),x)
Output:
Integral(csc(x)**3/(sin(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (38) = 76\).
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.31 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=-\frac {\frac {4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a} + \frac {\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {20 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{8 \, {\left (\frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {3 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a} \] Input:
integrate(csc(x)^3/(a+a*sin(x)),x, algorithm="maxima")
Output:
-1/8*(4*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/a + 1/8*(3*sin(x)/( cos(x) + 1) + 20*sin(x)^2/(cos(x) + 1)^2 - 1)/(a*sin(x)^2/(cos(x) + 1)^2 + a*sin(x)^3/(cos(x) + 1)^3) + 3/2*log(sin(x)/(cos(x) + 1))/a
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a} + \frac {a \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{2}} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a \tan \left (\frac {1}{2} \, x\right )^{2}} \] Input:
integrate(csc(x)^3/(a+a*sin(x)),x, algorithm="giac")
Output:
3/2*log(abs(tan(1/2*x)))/a + 1/8*(a*tan(1/2*x)^2 - 4*a*tan(1/2*x))/a^2 + 2 /(a*(tan(1/2*x) + 1)) - 1/8*(18*tan(1/2*x)^2 - 4*tan(1/2*x) + 1)/(a*tan(1/ 2*x)^2)
Time = 17.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{4\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a} \] Input:
int(1/(sin(x)^3*(a + a*sin(x))),x)
Output:
((3*tan(x/2))/2 + 10*tan(x/2)^2 - 1/2)/(4*a*tan(x/2)^2 + 4*a*tan(x/2)^3) - tan(x/2)/(2*a) + tan(x/2)^2/(8*a) + (3*log(tan(x/2)))/(2*a)
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.79 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {12 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{3}+12 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{2}+\tan \left (\frac {x}{2}\right )^{5}-3 \tan \left (\frac {x}{2}\right )^{4}-24 \tan \left (\frac {x}{2}\right )^{3}+3 \tan \left (\frac {x}{2}\right )-1}{8 \tan \left (\frac {x}{2}\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )} \] Input:
int(csc(x)^3/(a+a*sin(x)),x)
Output:
(12*log(tan(x/2))*tan(x/2)**3 + 12*log(tan(x/2))*tan(x/2)**2 + tan(x/2)**5 - 3*tan(x/2)**4 - 24*tan(x/2)**3 + 3*tan(x/2) - 1)/(8*tan(x/2)**2*a*(tan( x/2) + 1))