\(\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [77]
Optimal result
Integrand size = 21, antiderivative size = 21 \[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=i \text {Int}\left (-\frac {i \text {csch}(c+d x)}{a+b \tanh ^3(c+d x)},x\right )
\]
[Out]
I*Unintegrable(-I*csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)
Rubi [N/A]
Not integrable
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of
steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx
\]
[In]
Int[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]
[Out]
I*Defer[Int][((-I)*Csch[c + d*x])/(a + b*Tanh[c + d*x]^3), x]
Rubi steps \begin{align*}
\text {integral}& = i \int -\frac {i \text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(31)=62\).
Time = 0.63 (sec) , antiderivative size = 331, normalized size of antiderivative = 15.76
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {6 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ]}{6 a d}
\]
[In]
Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]
[Out]
-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] + Cos
h[(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)
Maple [N/A] (verified)
Time = 0.63 (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 |
\(-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (46656 a^{8} d^{6}-46656 a^{6} b^{2} d^{6}\right ) \textit {\_Z}^{6}+3888 a^{4} b^{2} d^{4} \textit {\_Z}^{4}-108 a^{2} b^{2} d^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {7776 a^{6} d^{5}}{b}-7776 b \,d^{5} a^{4}\right ) \textit {\_R}^{5}+\left (\frac {216 d^{3} a^{4}}{b}+432 b \,d^{3} a^{2}\right ) \textit {\_R}^{3}+\left (6 d a -6 b d \right ) \textit {\_R} \right )\right )+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}\) |
\(168\) |
| | |
|
|
|
|
[In]
int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
[Out]
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)))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.73 (sec) , antiderivative size = 20085, normalized size of antiderivative = 956.43
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
[Out]
Too large to include
Sympy [N/A]
Not integrable
Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx
\]
[In]
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)**3),x)
[Out]
Integral(csch(c + d*x)/(a + b*tanh(c + d*x)**3), x)
Maxima [N/A]
Not integrable
Time = 0.40 (sec) , antiderivative size = 160, normalized size of antiderivative = 7.62
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x }
\]
[In]
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
[Out]
-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)
Giac [N/A]
Not integrable
Time = 2.97 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x }
\]
[In]
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="giac")
[Out]
sage0*x
Mupad [B] (verification not implemented)
Time = 15.27 (sec) , antiderivative size = 3679, normalized size of antiderivative = 175.19
\[
\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^3)),x)
[Out]
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)) + 134217728*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 + 1550214758
4*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 + 18119393280*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 - 48922361856*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^8*b^3*d^5 - 32614907904*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)^7*a^6*b^7*d^7 -
179381993472*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)^7*a^7*b^6*d^7 - 16307453952*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)^7*a^8*b^5*d^7 + 179381993472*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)^7*a^9*b^4*d^7 + 48922361856*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)^7*a^10*b^3*d^7 - 1912602624*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^2*b^5*d - 100663296*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^3*b^4*d + 738197504*a*b^5
*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)) + 268435456*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)^2*a*b^7*d^2*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)) - 29158801408*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)^2*a^2*b^6*d^2*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)) - 29125246976*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)^2*a^3*b^5*d^2*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)) - 2113929216*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)^2*a^4*b^4*d^2*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)) - 4831838208*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)^4*a^3*b^7*d^4*exp(d*x)*exp(ro
ot(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)) + 1654904586
24*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)^4*a^4*b^
6*d^4*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)) + 283870494720*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)^4*a^5*b^5*d^4*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)) + 132573560832*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)^4*a^6*b^4*d^4*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)) + 2717908992*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 2
43*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^7*b^3*d^4*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)) + 21743271936*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)^6*a^5*b^7*d^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)) - 154920812544*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)^6*a^6*b^6*d^6*ex
p(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))
- 279944626176*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)^6*a^7*b^5*d^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)) - 105998450688*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)^6*a^8*b^4*d^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)) - 2717908992*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)^6*a^9*b^3*d^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)))/(13*a*b^12 + 13*a^12*b + a^13 + b^13 + 78*a^2*b
^11 + 286*a^3*b^10 + 715*a^4*b^9 + 1287*a^5*b^8 + 1716*a^6*b^7 + 1716*a^7*b^6 + 1287*a^8*b^5 + 715*a^9*b^4 + 2
86*a^10*b^3 + 78*a^11*b^2))*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), k, 1, 6) + log(exp(d*x + 1/(a*d)) - 1)/(a*d) - log(exp(d*x - 1/(a*d)) + 1)/(a*d)