Integrand size = 23, antiderivative size = 304 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=-\frac {2 b^{2/3} \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{2/3} \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {2 b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x)}{a d} \]
-coth(d*x+c)/a/d+2/3*(-1)^(1/3)*b^(2/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1 /3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2))/a^( 4/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)-2/3*b^(2/3)*arctanh((b^(1/3)-a^( 1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/a^(4/3)/d/(a^(2/3)+b^(2 /3))^(1/2)-2/3*b^(2/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tan h(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(4/3)/d /((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.80 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.76 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=-\frac {3 \coth \left (\frac {1}{2} (c+d x)\right )+2 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+c \text {$\#$1}^2+d x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \tanh \left (\frac {1}{2} (c+d x)\right )}{6 a d} \]
-1/6*(3*Coth[(c + d*x)/2] + 2*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^ 4 + b*#1^6 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[- Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d *x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*Tanh[(c + d*x)/2 ])/(a*d)
Time = 0.76 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 25, 3699, 2009}
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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a+i b \sin (i c+i d x)^3\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i c+i d x)^2 \left (i b \sin (i c+i d x)^3+a\right )}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -\int \left (\frac {b \sinh (c+d x)}{a \left (b \sinh ^3(c+d x)+a\right )}-\frac {\text {csch}^2(c+d x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^{2/3} \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{2/3} \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {2 b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}+b^{2/3}}}-\frac {\coth (c+d x)}{a d}\) |
(-2*b^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d *x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[( -1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*(-1)^(1/3)*b^(2/3)*ArcTan[ ((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^ (1/3)*a^(2/3) - b^(2/3)]])/(3*a^(4/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d ) - (2*b^(2/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + b^(2/3)]*d) - Coth[c + d*x]/(a*d)
3.2.79.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.57 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.39
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) | \(118\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) | \(118\) |
| risch | \(-\frac {2}{a d \left ({\mathrm e}^{2 d x +2 c}-1\right )}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (2985984 a^{10} d^{6}+2985984 a^{8} b^{2} d^{6}\right ) \textit {\_Z}^{6}+62208 a^{6} b^{2} d^{4} \textit {\_Z}^{4}-b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-\frac {248832 d^{5} a^{10}}{a^{2} b^{3}-b^{5}}-\frac {248832 d^{5} a^{8} b^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{5}+\left (\frac {20736 d^{4} a^{9}}{a^{2} b^{3}-b^{5}}+\frac {20736 d^{4} a^{7} b^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{4}+\left (-\frac {3456 d^{3} a^{6} b^{2}}{a^{2} b^{3}-b^{5}}+\frac {1728 d^{3} b^{4} a^{4}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{3}+\left (\frac {288 d^{2} a^{5} b^{2}}{a^{2} b^{3}-b^{5}}-\frac {144 d^{2} b^{4} a^{3}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{2}+\left (-\frac {12 d \,a^{4} b^{2}}{a^{2} b^{3}-b^{5}}+\frac {24 d \,b^{4} a^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R} -\frac {a \,b^{4}}{a^{2} b^{3}-b^{5}}\right )\right )\) | \(377\) |
1/d*(-1/2/a*tanh(1/2*d*x+1/2*c)-1/2/a/tanh(1/2*d*x+1/2*c)-2/3*b/a*sum((_R^ 3-_R)/(_R^5*a-2*_R^3*a-4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf (_Z^6*a-3*_Z^4*a-8*_Z^3*b+3*_Z^2*a-a)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 21133 vs. \(2 (211) = 422\).
Time = 1.05 (sec) , antiderivative size = 21133, normalized size of antiderivative = 69.52 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
-2/(a*d*e^(2*d*x + 2*c) - a*d) - 4*integrate((b*e^(4*d*x + 4*c) - b*e^(2*d *x + 2*c))/(a*b*e^(6*d*x + 6*c) - 3*a*b*e^(4*d*x + 4*c) + 8*a^2*e^(3*d*x + 3*c) + 3*a*b*e^(2*d*x + 2*c) - a*b), x)
\[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
Time = 24.14 (sec) , antiderivative size = 1293, normalized size of antiderivative = 4.25 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\left (\sum _{k=1}^6\ln \left (-\frac {8192\,b^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}-65536\,a\,b^3-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^3\,b^3\,d^2\,294912-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^4\,b^3\,d^3\,221184-\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^2\,b^3\,d\,196608+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^8\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,10616832+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^9\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7962624-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^6\,b\,d^3\,1769472+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^7\,b\,d^4\,2654208-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^8\,b\,d^5\,1990656+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^4\,b^2\,d^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,2064384+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^5\,b^2\,d^3\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,5529600+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^6\,b^2\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7299072+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^7\,b^2\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,9953280+\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^3\,b^2\,d\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,393216}{a^4\,b^5}\right )\,\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\right )+\frac {2}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}} \]
symsum(log(-(8192*b^4*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 24 3*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) - 65536*a*b^3 - 294912*root(729*a^8* b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^2*a^3*b^ 3*d^2 - 221184*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d ^4*z^4 - b^4, z, k)^3*a^4*b^3*d^3 - 196608*root(729*a^8*b^2*d^6*z^6 + 729* a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)*a^2*b^3*d + 10616832*root( 729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^ 4*a^8*d^4*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^ 4*z^4 - b^4, z, k) + d*x) + 7962624*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^ 6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^5*a^9*d^5*exp(root(729*a^8*b^2*d^ 6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) - 17694 72*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4 , z, k)^3*a^6*b*d^3 + 2654208*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^4*a^7*b*d^4 - 1990656*root(729*a^8*b^2* d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^5*a^8*b*d^5 + 2064384*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^ 4 - b^4, z, k)^2*a^4*b^2*d^2*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z ^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) + 5529600*root(729*a^8*b^2*d^ 6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^3*a^5*b^2*d^3* exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 -...