Integrand size = 21, antiderivative size = 21 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=i \text {Int}\left (-\frac {i \text {csch}(c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \] Output:
I*Defer(Int)(-I*csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)
Leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(31)=62\).
Time = 0.37 (sec) , antiderivative size = 331, normalized size of antiderivative = 15.76 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {6 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ]}{6 a d} \] Input:
Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]
Output:
-1/6*(6*Log[Cosh[(c + d*x)/2]] - 6*Log[Sinh[(c + d*x)/2]] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + 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] - 2*c*#1^2 - 2*d*x*#1^2 - 4*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 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b* #1^3 + a*#1^5 + b*#1^5) & ])/(a*d)
Not integrable
Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
\(\displaystyle \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a+i b \tan (i c+i d x)^3\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (i b \tan (i c+i d x)^3+a\right )}dx\) |
\(\Big \downarrow \) 4151 |
\(\displaystyle i \int -\frac {i \text {csch}(c+d x)}{b \tanh ^3(c+d x)+a}dx\) |
Input:
Int[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]
Output:
$Aborted
Time = 1.74 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.57
method | result | size |
derivativedivides | \(\frac {-\frac {4 b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(96\) |
default | \(\frac {-\frac {4 b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(96\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (46656 a^{8} d^{6}-46656 a^{6} b^{2} d^{6}\right ) \textit {\_Z}^{6}+3888 a^{4} b^{2} d^{4} \textit {\_Z}^{4}-108 a^{2} d^{2} \textit {\_Z}^{2} b^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {7776 d^{5} a^{6}}{b}-7776 b \,d^{5} a^{4}\right ) \textit {\_R}^{5}+\left (\frac {216 d^{3} a^{4}}{b}+432 a^{2} b \,d^{3}\right ) \textit {\_R}^{3}+\left (6 a d -6 b d \right ) \textit {\_R} \right )\right )+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}\) | \(168\) |
Input:
int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(-4/3*b/a*sum(_R^2/(_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))+1/a*ln(tanh(1/2*d* x+1/2*c)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.79 (sec) , antiderivative size = 20085, normalized size of antiderivative = 956.43 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
Output:
Too large to include
Not integrable
Time = 0.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)**3),x)
Output:
Integral(csch(c + d*x)/(a + b*tanh(c + d*x)**3), x)
Not integrable
Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 7.62 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
Output:
-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(5*d*x + 5*c) - 2*b*e^(3*d*x + 3*c) + b*e^(d*x + c))/( a^2 - a*b + (a^2*e^(6*c) + a*b*e^(6*c))*e^(6*d*x) + 3*(a^2*e^(4*c) - a*b*e ^(4*c))*e^(4*d*x) + 3*(a^2*e^(2*c) + a*b*e^(2*c))*e^(2*d*x)), x)
Not integrable
Time = 1.38 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="giac")
Output:
sage0*x
Time = 15.58 (sec) , antiderivative size = 3679, normalized size of antiderivative = 175.19 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \] Input:
int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^3)),x)
Output:
symsum(log(-(1409286144*b^6*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^ 8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 13421 7728*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27 *a^2*b^2*d^2*z^2 - b^2, z, k)*b^7*d + 1879048192*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)* a*b^6*d - 2818572288*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4* b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^7*d^3 - 40869298176* root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2* b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^6*d^3 + 28185722880*root(729*a^6*b^2*d^6* z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^5*d^3 + 15502147584*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*b^4*d^3 + 1 8119393280*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^ 4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^7*d^5 + 235552112640*root(729* a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z ^2 - b^2, z, k)^5*a^5*b^6*d^5 + 14495514624*root(729*a^6*b^2*d^6*z^6 - 729 *a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6 *b^5*d^5 - 219244658688*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a ^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^4*d^5 - 489223618 56*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 2...
Not integrable
Time = 0.32 (sec) , antiderivative size = 784, normalized size of antiderivative = 37.33 \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {12 e^{5 c} \left (\int \frac {e^{5 d x}}{e^{8 d x +8 c} a^{2}+2 e^{8 d x +8 c} a b +e^{8 d x +8 c} b^{2}+2 e^{6 d x +6 c} a^{2}-2 e^{6 d x +6 c} a b -4 e^{6 d x +6 c} b^{2}+6 e^{4 d x +4 c} a b +6 e^{4 d x +4 c} b^{2}-2 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b -4 e^{2 d x +2 c} b^{2}-a^{2}+b^{2}}d x \right ) a b d +12 e^{5 c} \left (\int \frac {e^{5 d x}}{e^{8 d x +8 c} a^{2}+2 e^{8 d x +8 c} a b +e^{8 d x +8 c} b^{2}+2 e^{6 d x +6 c} a^{2}-2 e^{6 d x +6 c} a b -4 e^{6 d x +6 c} b^{2}+6 e^{4 d x +4 c} a b +6 e^{4 d x +4 c} b^{2}-2 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b -4 e^{2 d x +2 c} b^{2}-a^{2}+b^{2}}d x \right ) b^{2} d +4 e^{c} \left (\int \frac {e^{d x}}{e^{8 d x +8 c} a^{2}+2 e^{8 d x +8 c} a b +e^{8 d x +8 c} b^{2}+2 e^{6 d x +6 c} a^{2}-2 e^{6 d x +6 c} a b -4 e^{6 d x +6 c} b^{2}+6 e^{4 d x +4 c} a b +6 e^{4 d x +4 c} b^{2}-2 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b -4 e^{2 d x +2 c} b^{2}-a^{2}+b^{2}}d x \right ) a b d +4 e^{c} \left (\int \frac {e^{d x}}{e^{8 d x +8 c} a^{2}+2 e^{8 d x +8 c} a b +e^{8 d x +8 c} b^{2}+2 e^{6 d x +6 c} a^{2}-2 e^{6 d x +6 c} a b -4 e^{6 d x +6 c} b^{2}+6 e^{4 d x +4 c} a b +6 e^{4 d x +4 c} b^{2}-2 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b -4 e^{2 d x +2 c} b^{2}-a^{2}+b^{2}}d x \right ) b^{2} d +\mathrm {log}\left (e^{d x +c}-1\right )-\mathrm {log}\left (e^{d x +c}+1\right )}{d \left (a +b \right )} \] Input:
int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)
Output:
(12*e**(5*c)*int(e**(5*d*x)/(e**(8*c + 8*d*x)*a**2 + 2*e**(8*c + 8*d*x)*a* b + e**(8*c + 8*d*x)*b**2 + 2*e**(6*c + 6*d*x)*a**2 - 2*e**(6*c + 6*d*x)*a *b - 4*e**(6*c + 6*d*x)*b**2 + 6*e**(4*c + 4*d*x)*a*b + 6*e**(4*c + 4*d*x) *b**2 - 2*e**(2*c + 2*d*x)*a**2 - 6*e**(2*c + 2*d*x)*a*b - 4*e**(2*c + 2*d *x)*b**2 - a**2 + b**2),x)*a*b*d + 12*e**(5*c)*int(e**(5*d*x)/(e**(8*c + 8 *d*x)*a**2 + 2*e**(8*c + 8*d*x)*a*b + e**(8*c + 8*d*x)*b**2 + 2*e**(6*c + 6*d*x)*a**2 - 2*e**(6*c + 6*d*x)*a*b - 4*e**(6*c + 6*d*x)*b**2 + 6*e**(4*c + 4*d*x)*a*b + 6*e**(4*c + 4*d*x)*b**2 - 2*e**(2*c + 2*d*x)*a**2 - 6*e**( 2*c + 2*d*x)*a*b - 4*e**(2*c + 2*d*x)*b**2 - a**2 + b**2),x)*b**2*d + 4*e* *c*int(e**(d*x)/(e**(8*c + 8*d*x)*a**2 + 2*e**(8*c + 8*d*x)*a*b + e**(8*c + 8*d*x)*b**2 + 2*e**(6*c + 6*d*x)*a**2 - 2*e**(6*c + 6*d*x)*a*b - 4*e**(6 *c + 6*d*x)*b**2 + 6*e**(4*c + 4*d*x)*a*b + 6*e**(4*c + 4*d*x)*b**2 - 2*e* *(2*c + 2*d*x)*a**2 - 6*e**(2*c + 2*d*x)*a*b - 4*e**(2*c + 2*d*x)*b**2 - a **2 + b**2),x)*a*b*d + 4*e**c*int(e**(d*x)/(e**(8*c + 8*d*x)*a**2 + 2*e**( 8*c + 8*d*x)*a*b + e**(8*c + 8*d*x)*b**2 + 2*e**(6*c + 6*d*x)*a**2 - 2*e** (6*c + 6*d*x)*a*b - 4*e**(6*c + 6*d*x)*b**2 + 6*e**(4*c + 4*d*x)*a*b + 6*e **(4*c + 4*d*x)*b**2 - 2*e**(2*c + 2*d*x)*a**2 - 6*e**(2*c + 2*d*x)*a*b - 4*e**(2*c + 2*d*x)*b**2 - a**2 + b**2),x)*b**2*d + log(e**(c + d*x) - 1) - log(e**(c + d*x) + 1))/(d*(a + b))