Integrand size = 23, antiderivative size = 322 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 (-1)^{2/3} b \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac {\text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {2 b \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
1/2*arctanh(cosh(d*x+c))/a/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d+2/3*(-1)^(2/3 )*b*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^(5/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^ (1/2)+2/3*b*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3) )^(1/2))/a^(5/3)/d/(a^(2/3)+b^(2/3))^(1/2)+2/3*(-1)^(2/3)*b*arctan((-1)^(1 /6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3) -(-1)^(2/3)*b^(2/3))^(1/2))/a^(5/3)/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 = 0.90 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.59 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=-\frac {16 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \]
-1/24*(16*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 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)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3 *(Csch[(c + d*x)/2]^2 - 4*Log[Cosh[(c + d*x)/2]] + 4*Log[Sinh[(c + d*x)/2] ] + Sech[(c + d*x)/2]^2))/(a*d)
Time = 0.70 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 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.
\(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a+i b \sin (i c+i d x)^3\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (i b \sin (i c+i d x)^3+a\right )}dx\) |
\(\Big \downarrow \) 3699 |
\(\displaystyle -i \int \left (\frac {i \text {csch}^3(c+d x)}{a}-\frac {i b}{a \left (b \sinh ^3(c+d x)+a\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i \left (-\frac {2 \sqrt [6]{-1} b \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 \sqrt [6]{-1} b \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 i b \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {i \text {arctanh}(\cosh (c+d x))}{2 a d}-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 a d}\right )\) |
(-I)*((-2*(-1)^(1/6)*b*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^( 5/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - (2*(-1)^(1/6)*b*Ar cTan[((-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^(5/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/ 3)]*d) + ((I/2)*ArcTanh[Cosh[c + d*x]])/(a*d) + (((2*I)/3)*b*ArcTanh[(b^(1 /3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b^(2/3)]])/(a^(5/3)*Sqrt[a ^(2/3) + b^(2/3)]*d) - ((I/2)*Coth[c + d*x]*Csch[c + d*x])/(a*d))
3.2.80.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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 = 0.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.43
method | result | size |
derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) | \(138\) |
default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) | \(138\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (191102976 a^{12} d^{6}+191102976 a^{10} b^{2} d^{6}\right ) \textit {\_Z}^{6}-995328 a^{8} b^{2} d^{4} \textit {\_Z}^{4}+1728 a^{4} b^{4} d^{2} \textit {\_Z}^{2}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {15925248 d^{5} a^{11}}{b^{6}}+\frac {15925248 d^{5} a^{9}}{b^{4}}\right ) \textit {\_R}^{5}+\left (\frac {331776 d^{4} a^{9}}{b^{5}}+\frac {331776 d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\left (-\frac {69120 d^{3} a^{7}}{b^{4}}+\frac {13824 d^{3} a^{5}}{b^{2}}\right ) \textit {\_R}^{3}-\frac {1728 d^{2} a^{5} \textit {\_R}^{2}}{b^{3}}+\frac {72 d \,a^{3} \textit {\_R}}{b^{2}}+\frac {2 a}{b}\right )\right )-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}\) | \(259\) |
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/a*ln(tanh (1/2*d*x+1/2*c))+1/3*b/a*sum((_R^4-2*_R^2+1)/(_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. 29179 vs. \(2 (229) = 458\).
Time = 7.76 (sec) , antiderivative size = 29179, normalized size of antiderivative = 90.62 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
-8*b*integrate(e^(3*d*x + 3*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) - (e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x + 4*c) - 2*a*d*e^(2*d*x + 2*c) + a*d) + 1/2*log((e^(d*x + c) + 1)*e^(-c))/(a*d) - 1/2*log((e^(d*x + c) - 1)*e^(-c ))/(a*d)
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
Time = 93.18 (sec) , antiderivative size = 3605, normalized size of antiderivative = 11.20 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Too large to display} \]
symsum(log(-(16777216*b^7*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^1 2*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) - 50331 648*a*b^6 + 33554432*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^ 8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)*a*b^7*d + 671088640*a^2*b^ 5*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2* d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 201326592*root(729*a^10*b^2*d ^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6 , z, k)^2*a^3*b^6*d^2 - 1509949440*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^ 6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^2*a^5*b^4*d^ 2 - 2717908992*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2* d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^3*a^5*b^5*d^3 + 2717908992*root( 729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4 *d^2*z^2 - b^6, z, k)^3*a^7*b^3*d^3 + 6039797760*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k )^4*a^7*b^4*d^4 - 4076863488*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^4*a^9*b^2*d^4 - 67 9477248*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^5*a^9*b^3*d^5 + 16307453952*root(729*a^ 10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z ^2 - b^6, z, k)^6*a^11*b^2*d^6 - 32614907904*root(729*a^10*b^2*d^6*z^6 ...