Integrand size = 14, antiderivative size = 109 \[ \int (c+d x) \csc ^3(a+b x) \, dx=-\frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2} \]
-(d*x+c)*arctanh(exp(I*(b*x+a)))/b-1/2*d*csc(b*x+a)/b^2-1/2*(d*x+c)*cot(b* x+a)*csc(b*x+a)/b+1/2*I*d*polylog(2,-exp(I*(b*x+a)))/b^2-1/2*I*d*polylog(2 ,exp(I*(b*x+a)))/b^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(292\) vs. \(2(109)=218\).
Time = 1.44 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.68 \[ \int (c+d x) \csc ^3(a+b x) \, dx=-\frac {d x \csc ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {c \csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {c \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {c \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {d \left ((a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )-a \log \left (\tan \left (\frac {1}{2} (a+b x)\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )\right )\right )}{2 b^2}+\frac {d x \sec ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {c \sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {d \csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {b x}{2}\right )}{4 b^2}-\frac {d \sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {b x}{2}\right )}{4 b^2} \]
-1/8*(d*x*Csc[a/2 + (b*x)/2]^2)/b - (c*Csc[(a + b*x)/2]^2)/(8*b) - (c*Log[ Cos[(a + b*x)/2]])/(2*b) + (c*Log[Sin[(a + b*x)/2]])/(2*b) + (d*((a + b*x) *(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) - a*Log[Tan[(a + b* x)/2]] + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/ (2*b^2) + (d*x*Sec[a/2 + (b*x)/2]^2)/(8*b) + (c*Sec[(a + b*x)/2]^2)/(8*b) + (d*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) - (d*Sec[a/2]*Sec[a /2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2)
Time = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4673, 3042, 4671, 2715, 2838}
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.
| \(\displaystyle \int (c+d x) \csc ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x) \csc (a+b x)^3dx\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle \frac {1}{2} \int (c+d x) \csc (a+b x)dx-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int (c+d x) \csc (a+b x)dx-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {1}{2} \left (-\frac {d \int \log \left (1-e^{i (a+b x)}\right )dx}{b}+\frac {d \int \log \left (1+e^{i (a+b x)}\right )dx}{b}-\frac {2 (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {1}{2} \left (\frac {i d \int e^{-i (a+b x)} \log \left (1-e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i d \int e^{-i (a+b x)} \log \left (1+e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {2 (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}\right )-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}\) |
-1/2*(d*Csc[a + b*x])/b^2 - ((c + d*x)*Cot[a + b*x]*Csc[a + b*x])/(2*b) + ((-2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b + (I*d*PolyLog[2, -E^(I*(a + b* x))])/b^2 - (I*d*PolyLog[2, E^(I*(a + b*x))])/b^2)/2
3.1.35.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (93 ) = 186\).
Time = 0.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.26
| method | result | size |
| risch | \(\frac {d x b \,{\mathrm e}^{3 i \left (b x +a \right )}+c b \,{\mathrm e}^{3 i \left (b x +a \right )}+d x b \,{\mathrm e}^{i \left (b x +a \right )}+c b \,{\mathrm e}^{i \left (b x +a \right )}-i d \,{\mathrm e}^{3 i \left (b x +a \right )}+i d \,{\mathrm e}^{i \left (b x +a \right )}}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {c \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{2 b}-\frac {d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{2 b^{2}}+\frac {i d \,\operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{2 b}+\frac {d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{2 b^{2}}-\frac {i d \,\operatorname {Li}_{2}\left ({\mathrm e}^{i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {d a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}\) | \(246\) |
1/b^2/(exp(2*I*(b*x+a))-1)^2*(d*x*b*exp(3*I*(b*x+a))+c*b*exp(3*I*(b*x+a))+ d*x*b*exp(I*(b*x+a))+c*b*exp(I*(b*x+a))-I*d*exp(3*I*(b*x+a))+I*d*exp(I*(b* x+a)))-1/b*c*arctanh(exp(I*(b*x+a)))-1/2/b*d*ln(exp(I*(b*x+a))+1)*x-1/2/b^ 2*d*ln(exp(I*(b*x+a))+1)*a+1/2*I*d*polylog(2,-exp(I*(b*x+a)))/b^2+1/2/b*d* ln(1-exp(I*(b*x+a)))*x+1/2/b^2*d*ln(1-exp(I*(b*x+a)))*a-1/2*I*d*polylog(2, exp(I*(b*x+a)))/b^2+1/b^2*d*a*arctanh(exp(I*(b*x+a)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (89) = 178\).
Time = 0.34 (sec) , antiderivative size = 452, normalized size of antiderivative = 4.15 \[ \int (c+d x) \csc ^3(a+b x) \, dx=\frac {2 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) + {\left (-i \, d \cos \left (b x + a\right )^{2} + i \, d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (i \, d \cos \left (b x + a\right )^{2} - i \, d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (-i \, d \cos \left (b x + a\right )^{2} + i \, d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (i \, d \cos \left (b x + a\right )^{2} - i \, d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (b d x - {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b d x - {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + {\left ({\left (b c - a d\right )} \cos \left (b x + a\right )^{2} - b c + a d\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left ({\left (b c - a d\right )} \cos \left (b x + a\right )^{2} - b c + a d\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - {\left (b d x - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} + a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b d x - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} + a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + 2 \, d \sin \left (b x + a\right )}{4 \, {\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \]
1/4*(2*(b*d*x + b*c)*cos(b*x + a) + (-I*d*cos(b*x + a)^2 + I*d)*dilog(cos( b*x + a) + I*sin(b*x + a)) + (I*d*cos(b*x + a)^2 - I*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) + (-I*d*cos(b*x + a)^2 + I*d)*dilog(-cos(b*x + a) + I*s in(b*x + a)) + (I*d*cos(b*x + a)^2 - I*d)*dilog(-cos(b*x + a) - I*sin(b*x + a)) + (b*d*x - (b*d*x + b*c)*cos(b*x + a)^2 + b*c)*log(cos(b*x + a) + I* sin(b*x + a) + 1) + (b*d*x - (b*d*x + b*c)*cos(b*x + a)^2 + b*c)*log(cos(b *x + a) - I*sin(b*x + a) + 1) + ((b*c - a*d)*cos(b*x + a)^2 - b*c + a*d)*l og(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + ((b*c - a*d)*cos(b*x + a)^2 - b*c + a*d)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - (b*d *x - (b*d*x + a*d)*cos(b*x + a)^2 + a*d)*log(-cos(b*x + a) + I*sin(b*x + a ) + 1) - (b*d*x - (b*d*x + a*d)*cos(b*x + a)^2 + a*d)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) + 2*d*sin(b*x + a))/(b^2*cos(b*x + a)^2 - b^2)
\[ \int (c+d x) \csc ^3(a+b x) \, dx=\int \left (c + d x\right ) \csc ^{3}{\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (89) = 178\).
Time = 0.32 (sec) , antiderivative size = 763, normalized size of antiderivative = 7.00 \[ \int (c+d x) \csc ^3(a+b x) \, dx=-\frac {2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (-i \, b d x - i \, b c\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (i \, b d x + i \, b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left (b c \cos \left (4 \, b x + 4 \, a\right ) - 2 \, b c \cos \left (2 \, b x + 2 \, a\right ) + i \, b c \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, b c \sin \left (2 \, b x + 2 \, a\right ) + b c\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) - 1\right ) + 2 \, {\left (b d x \cos \left (4 \, b x + 4 \, a\right ) - 2 \, b d x \cos \left (2 \, b x + 2 \, a\right ) + i \, b d x \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, b d x \sin \left (2 \, b x + 2 \, a\right ) + b d x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 4 \, {\left (-i \, b d x - i \, b c - d\right )} \cos \left (3 \, b x + 3 \, a\right ) - 4 \, {\left (-i \, b d x - i \, b c + d\right )} \cos \left (b x + a\right ) - 2 \, {\left (d \cos \left (4 \, b x + 4 \, a\right ) - 2 \, d \cos \left (2 \, b x + 2 \, a\right ) + i \, d \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, d \sin \left (2 \, b x + 2 \, a\right ) + d\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 2 \, {\left (d \cos \left (4 \, b x + 4 \, a\right ) - 2 \, d \cos \left (2 \, b x + 2 \, a\right ) + i \, d \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, d \sin \left (2 \, b x + 2 \, a\right ) + d\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + {\left (-i \, b d x - i \, b c + {\left (-i \, b d x - i \, b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (-i \, b d x - i \, b c\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b d x + b c\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + {\left (i \, b d x + i \, b c + {\left (i \, b d x + i \, b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (i \, b d x + i \, b c\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (b d x + b c\right )} \sin \left (4 \, b x + 4 \, a\right ) + 2 \, {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 4 \, {\left (b d x + b c - i \, d\right )} \sin \left (3 \, b x + 3 \, a\right ) - 4 \, {\left (b d x + b c + i \, d\right )} \sin \left (b x + a\right )}{-4 i \, b^{2} \cos \left (4 \, b x + 4 \, a\right ) + 8 i \, b^{2} \cos \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} \sin \left (4 \, b x + 4 \, a\right ) - 8 \, b^{2} \sin \left (2 \, b x + 2 \, a\right ) - 4 i \, b^{2}} \]
-(2*(b*d*x + b*c + (b*d*x + b*c)*cos(4*b*x + 4*a) - 2*(b*d*x + b*c)*cos(2* b*x + 2*a) - (-I*b*d*x - I*b*c)*sin(4*b*x + 4*a) - 2*(I*b*d*x + I*b*c)*sin (2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 2*(b*c*cos(4*b*x + 4*a) - 2*b*c*cos(2*b*x + 2*a) + I*b*c*sin(4*b*x + 4*a) - 2*I*b*c*sin(2*b *x + 2*a) + b*c)*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 2*(b*d*x*cos(4* b*x + 4*a) - 2*b*d*x*cos(2*b*x + 2*a) + I*b*d*x*sin(4*b*x + 4*a) - 2*I*b*d *x*sin(2*b*x + 2*a) + b*d*x)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4* (-I*b*d*x - I*b*c - d)*cos(3*b*x + 3*a) - 4*(-I*b*d*x - I*b*c + d)*cos(b*x + a) - 2*(d*cos(4*b*x + 4*a) - 2*d*cos(2*b*x + 2*a) + I*d*sin(4*b*x + 4*a ) - 2*I*d*sin(2*b*x + 2*a) + d)*dilog(-e^(I*b*x + I*a)) + 2*(d*cos(4*b*x + 4*a) - 2*d*cos(2*b*x + 2*a) + I*d*sin(4*b*x + 4*a) - 2*I*d*sin(2*b*x + 2* a) + d)*dilog(e^(I*b*x + I*a)) + (-I*b*d*x - I*b*c + (-I*b*d*x - I*b*c)*co s(4*b*x + 4*a) - 2*(-I*b*d*x - I*b*c)*cos(2*b*x + 2*a) + (b*d*x + b*c)*sin (4*b*x + 4*a) - 2*(b*d*x + b*c)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin (b*x + a)^2 + 2*cos(b*x + a) + 1) + (I*b*d*x + I*b*c + (I*b*d*x + I*b*c)*c os(4*b*x + 4*a) - 2*(I*b*d*x + I*b*c)*cos(2*b*x + 2*a) - (b*d*x + b*c)*sin (4*b*x + 4*a) + 2*(b*d*x + b*c)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin (b*x + a)^2 - 2*cos(b*x + a) + 1) - 4*(b*d*x + b*c - I*d)*sin(3*b*x + 3*a) - 4*(b*d*x + b*c + I*d)*sin(b*x + a))/(-4*I*b^2*cos(4*b*x + 4*a) + 8*I*b^ 2*cos(2*b*x + 2*a) + 4*b^2*sin(4*b*x + 4*a) - 8*b^2*sin(2*b*x + 2*a) - ...
\[ \int (c+d x) \csc ^3(a+b x) \, dx=\int { {\left (d x + c\right )} \csc \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x) \csc ^3(a+b x) \, dx=\text {Hanged} \]