3.2.100 \(\int \frac {\tanh ^5(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx\) [200]

Optimal. Leaf size=135 \[ \frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \tanh ^2(x)}{2 \sqrt {c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{4 c^{3/2}}+\frac {\tanh ^{-1}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c} \]

[Out]

1/4*(b-2*c)*arctanh(1/2*(b+2*c*tanh(x)^2)/c^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/c^(3/2)+1/2*arctanh(1/2*(
2*a+b+(b+2*c)*tanh(x)^2)/(a+b+c)^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/(a+b+c)^(1/2)-1/2*(a+b*tanh(x)^2+c*t
anh(x)^4)^(1/2)/c

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Rubi [A]
time = 0.25, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3781, 1265, 1667, 857, 635, 212, 738} \begin {gather*} \frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \tanh ^2(x)}{2 \sqrt {c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{4 c^{3/2}}-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c}+\frac {\tanh ^{-1}\left (\frac {2 a+(b+2 c) \tanh ^2(x)+b}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^5/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

((b - 2*c)*ArcTanh[(b + 2*c*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/(4*c^(3/2)) + ArcTanh
[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c])
- Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4]/(2*c)

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1667

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m
+ q + 2*p + 1))), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {\tanh ^5(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx &=-\text {Subst}\left (\int \frac {x^5}{\left (1+x^2\right ) \sqrt {a-b x^2+c x^4}} \, dx,x,i \tanh (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )\right )\\ &=-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c}-\frac {\text {Subst}\left (\int \frac {\frac {b}{2}+\frac {1}{2} (b-2 c) x}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )}{2 c}\\ &=-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )-\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )}{4 c}\\ &=-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c}-\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {-b-2 c \tanh ^2(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 c}+\text {Subst}\left (\int \frac {1}{4 a+4 b+4 c-x^2} \, dx,x,\frac {2 a+b+(b+2 c) \tanh ^2(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )\\ &=\frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \tanh ^2(x)}{2 \sqrt {c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{4 c^{3/2}}+\frac {\tanh ^{-1}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{2 c}\\ \end {align*}

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Mathematica [A]
time = 0.96, size = 136, normalized size = 1.01 \begin {gather*} \frac {1}{4} \left (\frac {(-b+2 c) \tanh ^{-1}\left (\frac {-b-2 c \tanh ^2(x)}{2 \sqrt {c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{c^{3/2}}+\frac {2 \tanh ^{-1}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{\sqrt {a+b+c}}-\frac {2 \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}{c}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^5/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

(((-b + 2*c)*ArcTanh[(-b - 2*c*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/c^(3/2) + (2*ArcTa
nh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/Sqrt[a + b + c] -
(2*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])/c)/4

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Maple [A]
time = 1.22, size = 149, normalized size = 1.10

method	result	size
derivativedivides	\(-\frac {\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\tanh ^{2}\left (x \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}\right )}{4 c^{\frac {3}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\tanh ^{2}\left (x \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}\right )}{2 \sqrt {c}}+\frac {\arctanh \left (\frac {b \left (\tanh ^{2}\left (x \right )\right )+2 c \left (\tanh ^{2}\left (x \right )\right )+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b +c}}\)	\(149\)
default	\(-\frac {\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\tanh ^{2}\left (x \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}\right )}{4 c^{\frac {3}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\tanh ^{2}\left (x \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}\right )}{2 \sqrt {c}}+\frac {\arctanh \left (\frac {b \left (\tanh ^{2}\left (x \right )\right )+2 c \left (\tanh ^{2}\left (x \right )\right )+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )+c \left (\tanh ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b +c}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^5/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2)/c+1/4*b/c^(3/2)*ln((1/2*b+c*tanh(x)^2)/c^(1/2)+(a+b*tanh(x)^2+c*tanh(x)
^4)^(1/2))-1/2*ln((1/2*b+c*tanh(x)^2)/c^(1/2)+(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/c^(1/2)+1/2/(a+b+c)^(1/2)*arc
tanh(1/2*(b*tanh(x)^2+2*c*tanh(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)^5/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2075 vs. \(2 (111) = 222\).
time = 1.20, size = 8891, normalized size = 65.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="fricas")

[Out]

[-1/8*(((a*b + b^2 - (2*a + b)*c - 2*c^2)*cosh(x)^4 + 4*(a*b + b^2 - (2*a + b)*c - 2*c^2)*cosh(x)*sinh(x)^3 +
(a*b + b^2 - (2*a + b)*c - 2*c^2)*sinh(x)^4 + 2*(a*b + b^2 - (2*a + b)*c - 2*c^2)*cosh(x)^2 + 2*(3*(a*b + b^2
- (2*a + b)*c - 2*c^2)*cosh(x)^2 + a*b + b^2 - (2*a + b)*c - 2*c^2)*sinh(x)^2 + a*b + b^2 - (2*a + b)*c - 2*c^
2 + 4*((a*b + b^2 - (2*a + b)*c - 2*c^2)*cosh(x)^3 + (a*b + b^2 - (2*a + b)*c - 2*c^2)*cosh(x))*sinh(x))*sqrt(
c)*log(((b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^8 + 8*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)*sinh(x)^7 + (b^2 + 4
*(a + 2*b)*c + 8*c^2)*sinh(x)^8 + 4*(b^2 + 4*a*c - 8*c^2)*cosh(x)^6 + 4*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(
x)^2 + b^2 + 4*a*c - 8*c^2)*sinh(x)^6 + 8*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^3 + 3*(b^2 + 4*a*c - 8*c^2)
*cosh(x))*sinh(x)^5 + 2*(3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^4 + 2*(35*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cos
h(x)^4 + 30*(b^2 + 4*a*c - 8*c^2)*cosh(x)^2 + 3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*sinh(x)^4 + 8*(7*(b^2 + 4*(a +
2*b)*c + 8*c^2)*cosh(x)^5 + 10*(b^2 + 4*a*c - 8*c^2)*cosh(x)^3 + (3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x))*
sinh(x)^3 + 4*(b^2 + 4*a*c - 8*c^2)*cosh(x)^2 + 4*(7*(b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^6 + 15*(b^2 + 4*a*c
- 8*c^2)*cosh(x)^4 + 3*(3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^2 + b^2 + 4*a*c - 8*c^2)*sinh(x)^2 - 4*sqrt
(2)*((b + 2*c)*cosh(x)^4 + 4*(b + 2*c)*cosh(x)*sinh(x)^3 + (b + 2*c)*sinh(x)^4 + 2*(b - 2*c)*cosh(x)^2 + 2*(3*
(b + 2*c)*cosh(x)^2 + b - 2*c)*sinh(x)^2 + 4*((b + 2*c)*cosh(x)^3 + (b - 2*c)*cosh(x))*sinh(x) + b + 2*c)*sqrt
(c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 + 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 + 2
*a - 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(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)) + b^2 + 4*(a + 2*b)*c + 8*c^2 + 8*((b^2 + 4*(a + 2*b)*c + 8*c^2)*cosh(x)^7 + 3*(b^2 + 4*a*
c - 8*c^2)*cosh(x)^5 + (3*b^2 + 4*(3*a - 2*b)*c + 24*c^2)*cosh(x)^3 + (b^2 + 4*a*c - 8*c^2)*cosh(x))*sinh(x))/
(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 +
3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(
x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(co
sh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)) - 2*(c^2*cosh(x)^4 + 4*c^2*cosh(x)*sinh(x)^3 + c^
2*sinh(x)^4 + 2*c^2*cosh(x)^2 + 2*(3*c^2*cosh(x)^2 + c^2)*sinh(x)^2 + c^2 + 4*(c^2*cosh(x)^3 + c^2*cosh(x))*si
nh(x))*sqrt(a + b + c)*log(((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a +
b)*c + c^2)*cosh(x)*sinh(x)^7 + (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 + 4*(a^2 + a*b - b*c - c^2)*
cosh(x)^6 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2)*sinh(x)^6 + 8*(7*(
a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^3 + 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2
*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 + 30*(a^2 + a*
b - b*c - c^2)*cosh(x)^2 + 3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b
)*c + c^2)*cosh(x)^5 + 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x))*s
inh(x)^3 + 4*(a^2 + a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 + 15*(
a^2 + a*b - b*c - c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2)*
sinh(x)^2 + sqrt(2)*((a + b + c)*cosh(x)^4 + 4*(a + b + c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 + 2*(a -
c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 + a - c)*sinh(x)^2 + 4*((a + b + c)*cosh(x)^3 + (a - c)*cosh(x))*sin
h(x) + a + b + c)*sqrt(a + b + c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 + 4*(a - c)*cosh(x)^2 +
2*(3*(a + b + c)*cosh(x)^2 + 2*a - 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(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 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b +
b^2 + 2*(a + b)*c + c^2)*cosh(x)^7 + 3*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^
2)*cosh(x)^3 + (a^2 + a*b - b*c - c^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)) + 4*sqrt(2)*((a + b)*c + c^2)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c
)*sinh(x)^4 + 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 + 2*a - 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(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 + b)*c^2 + c^3)*co
sh(x)^4 + 4*((a + b)*c^2 + c^3)*cosh(x)*sinh(x)^3 + ((a + b)*c^2 + c^3)*sinh(x)^4 + (a + b)*c^2 + c^3 + 2*((a
+ b)*c^2 + c^3)*cosh(x)^2 + 2*((a + b)*c^2 + c^3 + 3*((a + b)*c^2 + c^3)*cosh(x)^2)*sinh(x)^2 + 4*(((a + b)*c^
2 + c^3)*cosh(x)^3 + ((a + b)*c^2 + c^3)*cosh(x))*sinh(x)), -1/8*(4*(c^2*cosh(x)^4 + 4*c^2*cosh(x)*sinh(x)^3 +
c^2*sinh(x)^4 + 2*c^2*cosh(x)^2 + 2*(3*c^2*cos...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )} + c \tanh ^{4}{\left (x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**5/(a+b*tanh(x)**2+c*tanh(x)**4)**(1/2),x)

[Out]

Integral(tanh(x)**5/sqrt(a + b*tanh(x)**2 + c*tanh(x)**4), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="giac")

[Out]

integrate(tanh(x)^5/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {tanh}\left (x\right )}^5}{\sqrt {c\,{\mathrm {tanh}\left (x\right )}^4+b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^5/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2),x)

[Out]

int(tanh(x)^5/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2), x)

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