\(\int \frac {\tanh (x)}{(a+b \tanh ^4(x))^{3/2}} \, dx\) [262]
Optimal result
Integrand size = 15, antiderivative size = 74 \[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}}
\]
[Out]
1/2*arctanh((a+b*tanh(x)^2)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))/(a+b)^(3/2)+1/2*(-a+b*tanh(x)^2)/a/(a+b)/(a+b*t
anh(x)^4)^(1/2)
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 1262, 755, 12, 739, 212}
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}}
\]
[In]
Int[Tanh[x]/(a + b*Tanh[x]^4)^(3/2),x]
[Out]
ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*(a + b)^(3/2)) - (a - b*Tanh[x]^2)/(2*a*(a +
b)*Sqrt[a + b*Tanh[x]^4])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 739
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 755
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 1262
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]
Rule 3751
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rubi steps \begin{align*}
\text {integral}& = \text {Subst}\left (\int \frac {x}{\left (1-x^2\right ) \left (a+b x^4\right )^{3/2}} \, dx,x,\tanh (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh ^2(x)\right ) \\ & = -\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}}+\frac {\text {Subst}\left (\int \frac {a}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{2 a (a+b)} \\ & = -\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{2 (a+b)} \\ & = -\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a-b \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)} \\ & = \frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a-b \tanh ^2(x)}{2 a (a+b) \sqrt {a+b \tanh ^4(x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{(a+b)^{3/2}}-\frac {a-b \tanh ^2(x)}{a (a+b) \sqrt {a+b \tanh ^4(x)}}\right )
\]
[In]
Integrate[Tanh[x]/(a + b*Tanh[x]^4)^(3/2),x]
[Out]
(ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(a + b)^(3/2) - (a - b*Tanh[x]^2)/(a*(a + b)*S
qrt[a + b*Tanh[x]^4]))/2
Maple [C] (verified)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.68 (sec) , antiderivative size = 431, normalized size of antiderivative = 5.82
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {b \left (\frac {\tanh \left (x \right )^{3}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )^{2}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\tanh \left (x \right )^{4}+\frac {a}{b}\right ) b}}-\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}}{2 \left (a +b \right )}+\frac {b \left (-\frac {\tanh \left (x \right )^{3}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )^{2}}{4 a \left (a +b \right )}-\frac {\tanh \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\tanh \left (x \right )^{4}+\frac {a}{b}\right ) b}}-\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}}{2 \left (a +b \right )}\) |
\(431\) |
| default |
\(\frac {b \left (\frac {\tanh \left (x \right )^{3}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )^{2}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\tanh \left (x \right )^{4}+\frac {a}{b}\right ) b}}-\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}}{2 \left (a +b \right )}+\frac {b \left (-\frac {\tanh \left (x \right )^{3}}{4 a \left (a +b \right )}+\frac {\tanh \left (x \right )^{2}}{4 a \left (a +b \right )}-\frac {\tanh \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\tanh \left (x \right )^{4}+\frac {a}{b}\right ) b}}-\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}}{2 \left (a +b \right )}\) |
\(431\) |
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[In]
int(tanh(x)/(a+b*tanh(x)^4)^(3/2),x,method=_RETURNVERBOSE)
[Out]
b*(1/4/a/(a+b)*tanh(x)^3+1/4/a/(a+b)*tanh(x)^2+1/4/a/(a+b)*tanh(x)-1/4/(a+b)/b)/((tanh(x)^4+a/b)*b)^(1/2)-1/2/
(a+b)*(-1/2/(a+b)^(1/2)*arctanh(1/2*(2*b*tanh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))-1/(I/a^(1/2)*b^(1/2
))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^(1/2)*Ell
ipticPi(tanh(x)*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1
/2)))+b*(-1/4/a/(a+b)*tanh(x)^3+1/4/a/(a+b)*tanh(x)^2-1/4/a/(a+b)*tanh(x)-1/4/(a+b)/b)/((tanh(x)^4+a/b)*b)^(1/
2)-1/2/(a+b)*(-1/2/(a+b)^(1/2)*arctanh(1/2*(2*b*tanh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))+1/(I/a^(1/2)
*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^(1
/2)*EllipticPi(tanh(x)*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1
/2))^(1/2)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1935 vs. \(2 (63) = 126\).
Time = 0.48 (sec) , antiderivative size = 3914, normalized size of antiderivative = 52.89
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(tanh(x)/(a+b*tanh(x)^4)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((a^2 + a*b)*cosh(x)^8 + 8*(a^2 + a*b)*cosh(x)*sinh(x)^7 + (a^2 + a*b)*sinh(x)^8 + 4*(a^2 - a*b)*cosh(x)
^6 + 4*(7*(a^2 + a*b)*cosh(x)^2 + a^2 - a*b)*sinh(x)^6 + 8*(7*(a^2 + a*b)*cosh(x)^3 + 3*(a^2 - a*b)*cosh(x))*s
inh(x)^5 + 6*(a^2 + a*b)*cosh(x)^4 + 2*(35*(a^2 + a*b)*cosh(x)^4 + 30*(a^2 - a*b)*cosh(x)^2 + 3*a^2 + 3*a*b)*s
inh(x)^4 + 8*(7*(a^2 + a*b)*cosh(x)^5 + 10*(a^2 - a*b)*cosh(x)^3 + 3*(a^2 + a*b)*cosh(x))*sinh(x)^3 + 4*(a^2 -
a*b)*cosh(x)^2 + 4*(7*(a^2 + a*b)*cosh(x)^6 + 15*(a^2 - a*b)*cosh(x)^4 + 9*(a^2 + a*b)*cosh(x)^2 + a^2 - a*b)
*sinh(x)^2 + a^2 + a*b + 8*((a^2 + a*b)*cosh(x)^7 + 3*(a^2 - a*b)*cosh(x)^5 + 3*(a^2 + a*b)*cosh(x)^3 + (a^2 -
a*b)*cosh(x))*sinh(x))*sqrt(a + b)*log(((a^2 + 2*a*b + b^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)
^7 + (a^2 + 2*a*b + b^2)*sinh(x)^8 + 4*(a^2 - b^2)*cosh(x)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^2 + a^2 - b^2)
*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 - b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 3*b^2)
*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 30*(a^2 - b^2)*cosh(x)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(x)^4
+ 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^5 + 10*(a^2 - b^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x))*sinh(x)^3
+ 4*(a^2 - b^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^6 + 15*(a^2 - b^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b +
3*b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + sqrt(2)*((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*
sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 + (a - b)*c
osh(x))*sinh(x) + a + b)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 + 4*(a - b)*cosh(x)^2 + 2*(3*
(a + b)*cosh(x)^2 + 2*a - 2*b)*sinh(x)^2 + 3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2
- 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + a^2 + 2*a*b + b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(x)^7 + 3*(a^2 - b^2)*co
sh(x)^5 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) +
6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) - 2*sqrt(2)*((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*
cosh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 + 2*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 + a^2
+ 2*a*b + b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 + (a^2 + 2*a*b + b^2)*cosh(x))*sinh(x))*sqrt(
((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 + 4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + 2*a - 2*b)*sinh(x)^2 +
3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4)))/((a^
4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^8 + 8*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + (a^4 +
3*a^3*b + 3*a^2*b^2 + a*b^3)*sinh(x)^8 + 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^6 + 4*(a^4 + a^3*b - a^2*b^
2 - a*b^3 + 7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b
^3)*cosh(x)^3 + 3*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x))*sinh(x)^5 + 6*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*c
osh(x)^4 + 2*(35*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^4 + 3*a^4 + 9*a^3*b + 9*a^2*b^2 + 3*a*b^3 + 30*(a
^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + 8*(7*(a^4 + 3*a^3*b +
3*a^2*b^2 + a*b^3)*cosh(x)^5 + 10*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^3 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*cosh(x))*sinh(x)^3 + 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2 + 4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*
b^3)*cosh(x)^6 + 15*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^4 + a^4 + a^3*b - a^2*b^2 - a*b^3 + 9*(a^4 + 3*a^3
*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^7 + 3*(a^4 + a^3
*b - a^2*b^2 - a*b^3)*cosh(x)^5 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^3 + (a^4 + a^3*b - a^2*b^2 - a
*b^3)*cosh(x))*sinh(x)), -1/2*(((a^2 + a*b)*cosh(x)^8 + 8*(a^2 + a*b)*cosh(x)*sinh(x)^7 + (a^2 + a*b)*sinh(x)^
8 + 4*(a^2 - a*b)*cosh(x)^6 + 4*(7*(a^2 + a*b)*cosh(x)^2 + a^2 - a*b)*sinh(x)^6 + 8*(7*(a^2 + a*b)*cosh(x)^3 +
3*(a^2 - a*b)*cosh(x))*sinh(x)^5 + 6*(a^2 + a*b)*cosh(x)^4 + 2*(35*(a^2 + a*b)*cosh(x)^4 + 30*(a^2 - a*b)*cos
h(x)^2 + 3*a^2 + 3*a*b)*sinh(x)^4 + 8*(7*(a^2 + a*b)*cosh(x)^5 + 10*(a^2 - a*b)*cosh(x)^3 + 3*(a^2 + a*b)*cosh
(x))*sinh(x)^3 + 4*(a^2 - a*b)*cosh(x)^2 + 4*(7*(a^2 + a*b)*cosh(x)^6 + 15*(a^2 - a*b)*cosh(x)^4 + 9*(a^2 + a*
b)*cosh(x)^2 + a^2 - a*b)*sinh(x)^2 + a^2 + a*b + 8*((a^2 + a*b)*cosh(x)^7 + 3*(a^2 - a*b)*cosh(x)^5 + 3*(a^2
+ a*b)*cosh(x)^3 + (a^2 - a*b)*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*((a + b)*cosh(x)^4 + 4*(a + b)*co
sh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*sinh(x)^2 + 4*((a
+ b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 +
4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + 2*a - 2*b)*sinh(x)^2 + 3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh
(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4))/((a^2 + 2*a*b + b^2)*cosh(x)^8 + 8*(a^2 + 2*a*
b + b^2)*cosh(x)*sinh(x)^7 + (a^2 + 2*a*b + b^2)*sinh(x)^8 + 4*(a^2 - b^2)*cosh(x)^6 + 4*(7*(a^2 + 2*a*b + b^2
)*cosh(x)^2 + a^2 - b^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 - b^2)*cosh(x))*sinh(x)^5 + 6
*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 30*(a^2 - b^2)*cosh(x)^2 + 3*a^2 + 6*a*
b + 3*b^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^5 + 10*(a^2 - b^2)*cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*c
osh(x))*sinh(x)^3 + 4*(a^2 - b^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^6 + 15*(a^2 - b^2)*cosh(x)^4 +
9*(a^2 + 2*a*b + b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(x)^7
+ 3*(a^2 - b^2)*cosh(x)^5 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))) + sqrt(2)*((a^2 -
b^2)*cosh(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 + 2*(a^2 + 2*a*b + b^2)*cosh(x)^2 +
2*(3*(a^2 - b^2)*cosh(x)^2 + a^2 + 2*a*b + b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 + (a^2 + 2*a*
b + b^2)*cosh(x))*sinh(x))*sqrt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 + 4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*co
sh(x)^2 + 2*a - 2*b)*sinh(x)^2 + 3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(
x)*sinh(x)^3 + sinh(x)^4)))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^8 + 8*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*
b^3)*cosh(x)*sinh(x)^7 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sinh(x)^8 + 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cos
h(x)^6 + 4*(a^4 + a^3*b - a^2*b^2 - a*b^3 + 7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*
(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^3 + 3*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x))*sinh(x)^5 + 6*(a^4
+ 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^4 + 2*(35*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^4 + 3*a^4 + 9*a^3
*b + 9*a^2*b^2 + 3*a*b^3 + 30*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 3*a^3*b + 3*a^2*b^2
+ a*b^3 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^5 + 10*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^3 +
3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x))*sinh(x)^3 + 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2 + 4*(7
*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^6 + 15*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^4 + a^4 + a^3*b -
a^2*b^2 - a*b^3 + 9*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*cosh(x)^7 + 3*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^5 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x
)^3 + (a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x))*sinh(x))]
Sympy [F]
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\int \frac {\tanh {\left (x \right )}}{\left (a + b \tanh ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx
\]
[In]
integrate(tanh(x)/(a+b*tanh(x)**4)**(3/2),x)
[Out]
Integral(tanh(x)/(a + b*tanh(x)**4)**(3/2), x)
Maxima [F]
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\int { \frac {\tanh \left (x\right )}{{\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate(tanh(x)/(a+b*tanh(x)^4)^(3/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)/(b*tanh(x)^4 + a)^(3/2), x)
Giac [F]
\[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\int { \frac {\tanh \left (x\right )}{{\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate(tanh(x)/(a+b*tanh(x)^4)^(3/2),x, algorithm="giac")
[Out]
integrate(tanh(x)/(b*tanh(x)^4 + a)^(3/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{3/2}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{{\left (b\,{\mathrm {tanh}\left (x\right )}^4+a\right )}^{3/2}} \,d x
\]
[In]
int(tanh(x)/(a + b*tanh(x)^4)^(3/2),x)
[Out]
int(tanh(x)/(a + b*tanh(x)^4)^(3/2), x)