3.194 \(\int \frac {\tanh ^5(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx\)
Optimal. Leaf size=66 \[ -\frac {\left (a+b \text {sech}^2(x)\right )^{3/2}}{3 b^2}+\frac {(a+2 b) \sqrt {a+b \text {sech}^2(x)}}{b^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
[Out]
-1/3*(a+b*sech(x)^2)^(3/2)/b^2+arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))/a^(1/2)+(a+2*b)*(a+b*sech(x)^2)^(1/2)/b^
2
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Rubi [A] time = 0.13, antiderivative size = 66, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used =
{4139, 446, 88, 63, 208} \[ -\frac {\left (a+b \text {sech}^2(x)\right )^{3/2}}{3 b^2}+\frac {(a+2 b) \sqrt {a+b \text {sech}^2(x)}}{b^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^5/Sqrt[a + b*Sech[x]^2],x]
[Out]
ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] + ((a + 2*b)*Sqrt[a + b*Sech[x]^2])/b^2 - (a + b*Sech[x]^2)^(3/
2)/(3*b^2)
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 88
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 4139
Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p)/x
, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[
n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x \sqrt {a+b x^2}} \, dx,x,\text {sech}(x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x)^2}{x \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-a-2 b}{b \sqrt {a+b x}}+\frac {1}{x \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,\text {sech}^2(x)\right )\right )\\ &=\frac {(a+2 b) \sqrt {a+b \text {sech}^2(x)}}{b^2}-\frac {\left (a+b \text {sech}^2(x)\right )^{3/2}}{3 b^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )\\ &=\frac {(a+2 b) \sqrt {a+b \text {sech}^2(x)}}{b^2}-\frac {\left (a+b \text {sech}^2(x)\right )^{3/2}}{3 b^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {sech}^2(x)}\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {(a+2 b) \sqrt {a+b \text {sech}^2(x)}}{b^2}-\frac {\left (a+b \text {sech}^2(x)\right )^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 109, normalized size = 1.65 \[ \frac {\text {sech}(x) \left (\frac {\text {sech}(x) (a \cosh (2 x)+a+2 b) \left (2 a-b \text {sech}^2(x)+6 b\right )}{3 b^2}+\frac {\sqrt {2} \sqrt {a \cosh (2 x)+a+2 b} \log \left (\sqrt {a \cosh (2 x)+a+2 b}+\sqrt {2} \sqrt {a} \cosh (x)\right )}{\sqrt {a}}\right )}{2 \sqrt {a+b \text {sech}^2(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^5/Sqrt[a + b*Sech[x]^2],x]
[Out]
(Sech[x]*((Sqrt[2]*Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/Sqr
t[a] + ((a + 2*b + a*Cosh[2*x])*Sech[x]*(2*a + 6*b - b*Sech[x]^2))/(3*b^2)))/(2*Sqrt[a + b*Sech[x]^2])
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fricas [B] time = 0.61, size = 2678, normalized size = 40.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^5/(a+b*sech(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/12*(3*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2
)*sinh(x)^4 + 3*b^2*cosh(x)^2 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 3*(5*b^2*cosh(x)^4 + 6*b^2*cos
h(x)^2 + b^2)*sinh(x)^2 + b^2 + 6*(b^2*cosh(x)^5 + 2*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a)*log(((a^3 +
2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^8
+ 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^6 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 + 14*(a^3 + 2*a^2*b + a*b
^2)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(
x))*sinh(x)^5 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)^4 + (70*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 6*a^3 + 14*a^
2*b + 9*a*b^2 + 30*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh
(x)^5 + 10*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^3 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3
+ 2*(2*a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 + 15*(2*a^3 + 5*a^2*b + 4*a*b^2 + b
^3)*cosh(x)^4 + 2*a^3 + 3*a^2*b + 3*(6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*((a^2 + 2*a*b
+ b^2)*cosh(x)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^5 + (a^2 + 2*a*b + b^2)*sinh(x)^6 + 3*(a^2 + 2*a*b +
b^2)*cosh(x)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(x)^2 + a^2 + 2*a*b + b^2)*sinh(x)^4 + 4*(5*(a^2 + 2*a*b + b^2)*
cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x))*sinh(x)^3 + (3*a^2 + 4*a*b)*cosh(x)^2 + (15*(a^2 + 2*a*b + b^2)*cos
h(x)^4 + 18*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 3*a^2 + 4*a*b)*sinh(x)^2 + a^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^
5 + 6*(a^2 + 2*a*b + b^2)*cosh(x)^3 + (3*a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^
2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^7 + 3*(2*a^3
+ 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^5 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)^3 + (2*a^3 + 3*a^2*b)*cosh(x))*sin
h(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x
)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*c
osh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 3*b^2*cosh(x)^2 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)
^3 + 3*(5*b^2*cosh(x)^4 + 6*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 6*(b^2*cosh(x)^5 + 2*b^2*cosh(x)^3 + b^2*co
sh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*a*cosh(
x)^2 + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a)*sqrt((a*cosh(x)^2 + a*si
nh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(a*cosh(x)^3 + b*cosh(x))*sinh(x) + a)/(co
sh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 8*sqrt(2)*((a^2 + 3*a*b)*cosh(x)^4 + 4*(a^2 + 3*a*b)*cosh(x)*sinh(
x)^3 + (a^2 + 3*a*b)*sinh(x)^4 + 2*(a^2 + 2*a*b)*cosh(x)^2 + 2*(3*(a^2 + 3*a*b)*cosh(x)^2 + a^2 + 2*a*b)*sinh(
x)^2 + a^2 + 3*a*b + 4*((a^2 + 3*a*b)*cosh(x)^3 + (a^2 + 2*a*b)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x
)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a*b^2*cosh(x)^6 + 6*a*b^2*cosh(x)*sinh(x)^5 + a*
b^2*sinh(x)^6 + 3*a*b^2*cosh(x)^4 + 3*a*b^2*cosh(x)^2 + 3*(5*a*b^2*cosh(x)^2 + a*b^2)*sinh(x)^4 + 4*(5*a*b^2*c
osh(x)^3 + 3*a*b^2*cosh(x))*sinh(x)^3 + a*b^2 + 3*(5*a*b^2*cosh(x)^4 + 6*a*b^2*cosh(x)^2 + a*b^2)*sinh(x)^2 +
6*(a*b^2*cosh(x)^5 + 2*a*b^2*cosh(x)^3 + a*b^2*cosh(x))*sinh(x)), -1/6*(3*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(
x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 3*b^2*cosh(x)^2 + 4*(5*b^2*cosh
(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 3*(5*b^2*cosh(x)^4 + 6*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 6*(b^2*cosh(x
)^5 + 2*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*s
inh(x) + (a + b)*sinh(x)^2 + a)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sin
h(x) + sinh(x)^2))/((a^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3 + (a^2 + a*b)*sinh(x)^4 + (2*a^2 +
3*a*b)*cosh(x)^2 + (6*(a^2 + a*b)*cosh(x)^2 + 2*a^2 + 3*a*b)*sinh(x)^2 + a^2 + 2*(2*(a^2 + a*b)*cosh(x)^3 + (
2*a^2 + 3*a*b)*cosh(x))*sinh(x))) + 3*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)
^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 3*b^2*cosh(x)^2 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 3
*(5*b^2*cosh(x)^4 + 6*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 6*(b^2*cosh(x)^5 + 2*b^2*cosh(x)^3 + b^2*cosh(x))
*sinh(x))*sqrt(-a)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a)*sqrt((a*cosh(x)^2 +
a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*
sinh(x)^4 + 2*(a + 2*b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + 2*b)*sinh(x)^2 + 4*(a*cosh(x)^3 + (a + 2*b)*cosh(x)
)*sinh(x) + a)) - 4*sqrt(2)*((a^2 + 3*a*b)*cosh(x)^4 + 4*(a^2 + 3*a*b)*cosh(x)*sinh(x)^3 + (a^2 + 3*a*b)*sinh(
x)^4 + 2*(a^2 + 2*a*b)*cosh(x)^2 + 2*(3*(a^2 + 3*a*b)*cosh(x)^2 + a^2 + 2*a*b)*sinh(x)^2 + a^2 + 3*a*b + 4*((a
^2 + 3*a*b)*cosh(x)^3 + (a^2 + 2*a*b)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2
- 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a*b^2*cosh(x)^6 + 6*a*b^2*cosh(x)*sinh(x)^5 + a*b^2*sinh(x)^6 + 3*a*b^2*co
sh(x)^4 + 3*a*b^2*cosh(x)^2 + 3*(5*a*b^2*cosh(x)^2 + a*b^2)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 + 3*a*b^2*cosh(x)
)*sinh(x)^3 + a*b^2 + 3*(5*a*b^2*cosh(x)^4 + 6*a*b^2*cosh(x)^2 + a*b^2)*sinh(x)^2 + 6*(a*b^2*cosh(x)^5 + 2*a*b
^2*cosh(x)^3 + a*b^2*cosh(x))*sinh(x))]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^5/(a+b*sech(x)^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{5}\relax (x )}{\sqrt {a +b \mathrm {sech}\relax (x )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^5/(a+b*sech(x)^2)^(1/2),x)
[Out]
int(tanh(x)^5/(a+b*sech(x)^2)^(1/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)^{5}}{\sqrt {b \operatorname {sech}\relax (x)^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^5/(a+b*sech(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^5/sqrt(b*sech(x)^2 + a), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {tanh}\relax (x)}^5}{\sqrt {a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^5/(a + b/cosh(x)^2)^(1/2),x)
[Out]
int(tanh(x)^5/(a + b/cosh(x)^2)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{5}{\relax (x )}}{\sqrt {a + b \operatorname {sech}^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**5/(a+b*sech(x)**2)**(1/2),x)
[Out]
Integral(tanh(x)**5/sqrt(a + b*sech(x)**2), x)
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