\(\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [79]
Optimal result
Integrand size = 23, antiderivative size = 23 \[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-i \text {Int}\left (\frac {i \text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right )
\]
[Out]
-I*Unintegrable(I*csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x)
Rubi [N/A]
Not integrable
Time = 0.04 (sec) , antiderivative size = 23, 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 = {} \[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx
\]
[In]
Int[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]
[Out]
(-I)*Defer[Int][(I*Csch[c + d*x]^3)/(a + b*Tanh[c + d*x]^3), x]
Rubi steps \begin{align*}
\text {integral}& = -\left (i \int \frac {i \text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\right ) \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(214\) vs. \(2(33)=66\).
Time = 0.92 (sec) , antiderivative size = 214, normalized size of antiderivative = 9.30
\[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {16 b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{a+b+2 a \text {$\#$1}^2-2 b \text {$\#$1}^2+a \text {$\#$1}^4+b \text {$\#$1}^4}\&\right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d}
\]
[In]
Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]
[Out]
-1/24*(16*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*#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)/(a + b + 2*a*#1
^2 - 2*b*#1^2 + a*#1^4 + 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)
Maple [N/A] (verified)
Time = 1.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.91
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) |
\(136\) |
| default |
\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) |
\(136\) |
| risch |
\(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (191102976 d^{6} \textit {\_Z}^{6} a^{10}+1728 a^{4} b^{2} d^{2} \textit {\_Z}^{2}+a^{2} b^{2}-b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\frac {15925248 d^{5} a^{9} \textit {\_R}^{5}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\left (-\frac {13824 d^{3} a^{7}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\frac {13824 d^{3} b^{2} a^{5}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {24 d \,a^{4} b}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\frac {96 d \,b^{2} a^{3}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}-\frac {24 d \,b^{3} a^{2}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}\right ) \textit {\_R} \right )\right )-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}\) |
\(329\) |
| | |
|
|
|
|
[In]
int(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
[Out]
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-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))-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/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 = 2.94 (sec) , antiderivative size = 6846, normalized size of antiderivative = 297.65
\[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
[Out]
Too large to include
Sympy [N/A]
Not integrable
Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
\[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx
\]
[In]
integrate(csch(d*x+c)**3/(a+b*tanh(d*x+c)**3),x)
[Out]
Integral(csch(c + d*x)**3/(a + b*tanh(c + d*x)**3), x)
Maxima [N/A]
Not integrable
Time = 0.45 (sec) , antiderivative size = 188, normalized size of antiderivative = 8.17
\[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{3} + a} \,d x }
\]
[In]
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
[Out]
-8*b*integrate(e^(3*d*x + 3*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) - (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)
Giac [F(-2)]
Exception generated. \[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Exception raised: AttributeError}
\]
[In]
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm="giac")
[Out]
Exception raised: AttributeError >> type
Mupad [B] (verification not implemented)
Time = 27.78 (sec) , antiderivative size = 3643, normalized size of antiderivative = 158.39
\[
\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)),x)
[Out]
exp(c + d*x)/(a*d - a*d*exp(2*c + 2*d*x)) - (2*exp(c + d*x))/(a*d - 2*a*d*exp(2*c + 2*d*x) + a*d*exp(4*c + 4*d
*x)) + symsum(log((570425344*a^4*b^6*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4,
z, k)) - 33554432*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a*b^10*d - 553648128*a^2*b
^8*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 167772160*a^3*b^7*exp(d*x
)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 16777216*b^10*exp(d*x)*exp(root(729
*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 192937984*a^5*b^5*exp(d*x)*exp(root(729*a^10*d^6*
z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2617245696*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2
*b^2 - b^4, z, k)^3*a^5*b^8*d^3 - 150994944*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^
3*a^6*b^7*d^3 - 1384120320*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^7*b^6*d^3 + 2
415919104*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^8*b^5*d^3 - 3498049536*root(72
9*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^9*b^4*d^3 + 5435817984*root(729*a^10*d^6*z^6 +
27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^8*b^7*d^5 + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2
+ a^2*b^2 - b^4, z, k)^5*a^9*b^6*d^5 - 70665633792*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4
, z, k)^5*a^10*b^5*d^5 + 52319748096*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^11*
b^4*d^5 + 12230590464*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^12*b^3*d^5 + 32614
907904*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^11*b^6*d^7 + 146767085568*root(72
9*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^12*b^5*d^7 - 130459631616*root(729*a^10*d^6*z^6
+ 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^13*b^4*d^7 - 48922361856*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d
^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^14*b^3*d^7 + 67108864*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 -
b^4, z, k)*a^2*b^9*d - 427819008*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^3*b^8*d
- 822083584*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^4*b^7*d + 436207616*root(729*a
^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^5*b^6*d + 754974720*root(729*a^10*d^6*z^6 + 27*a^4*b
^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^6*b^5*d + 25165824*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 -
b^4, z, k)*a^7*b^4*d - 25165824*a*b^9*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4,
z, k)) + 234881024*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^3*b^9*d^2*exp(d*x)*e
xp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2592079872*root(729*a^10*d^6*z^6 + 27*
a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^4*b^8*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)) - 2860515328*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^5*b^
7*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2919235584*root(729*a^
10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^6*b^6*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a
^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 2357198848*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4
, z, k)^2*a^7*b^5*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 528482
304*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^8*b^4*d^2*exp(d*x)*exp(root(729*a^10
*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 301989888*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)^4*a^6*b^8*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z
, k)) + 9965666304*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^7*b^7*d^4*exp(d*x)*ex
p(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 33671872512*root(729*a^10*d^6*z^6 + 27*
a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^8*b^6*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)) - 6568280064*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^9*b^
5*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 29293019136*root(729*a
^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^10*b^4*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27
*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^
4, z, k)^4*a^11*b^3*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 7202
4588288*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^6*a^10*b^6*d^6*exp(d*x)*exp(root(729
*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 27179089920*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^
2*z^2 + a^2*b^2 - b^4, z, k)^6*a^11*b^5*d^6*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2
- b^4, z, k)) - 96485769216*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^6*a^12*b^4*d^6*e
xp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 2717908992*root(729*a^10*d^6*
z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^6*a^13*b^3*d^6*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2
*d^2*z^2 + a^2*b^2 - b^4, z, k)))/(12*a^16*b + a^17 + a^5*b^12 + 12*a^6*b^11 + 66*a^7*b^10 + 220*a^8*b^9 + 495
*a^9*b^8 + 792*a^10*b^7 + 924*a^11*b^6 + 792*a^12*b^5 + 495*a^13*b^4 + 220*a^14*b^3 + 66*a^15*b^2))*root(729*a
^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k), k, 1, 6) - log(33554432*a*b^9 - 16777216*b^10 + 11324
6208*a^2*b^8 - 260046848*a^3*b^7 + 321126400*a^4*b^6 - 382205952*a^5*b^5 + 191102976*a^6*b^4 + 16777216*b^10*e
xp(-1/(2*a*d))*exp(d*x) - 33554432*a*b^9*exp(-1/(2*a*d))*exp(d*x) - 113246208*a^2*b^8*exp(-1/(2*a*d))*exp(d*x)
+ 260046848*a^3*b^7*exp(-1/(2*a*d))*exp(d*x) - 321126400*a^4*b^6*exp(-1/(2*a*d))*exp(d*x) + 382205952*a^5*b^5
*exp(-1/(2*a*d))*exp(d*x) - 191102976*a^6*b^4*exp(-1/(2*a*d))*exp(d*x))/(2*a*d) + log(33554432*a*b^9 - 1677721
6*b^10 + 113246208*a^2*b^8 - 260046848*a^3*b^7 + 321126400*a^4*b^6 - 382205952*a^5*b^5 + 191102976*a^6*b^4 - 1
6777216*b^10*exp(1/(2*a*d))*exp(d*x) + 33554432*a*b^9*exp(1/(2*a*d))*exp(d*x) + 113246208*a^2*b^8*exp(1/(2*a*d
))*exp(d*x) - 260046848*a^3*b^7*exp(1/(2*a*d))*exp(d*x) + 321126400*a^4*b^6*exp(1/(2*a*d))*exp(d*x) - 38220595
2*a^5*b^5*exp(1/(2*a*d))*exp(d*x) + 191102976*a^6*b^4*exp(1/(2*a*d))*exp(d*x))/(2*a*d)