3.196 \(\int \frac{\tanh ^3(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx\)
Optimal. Leaf size=42 \[ \frac{\sqrt{a+b \text{sech}^2(x)}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}} \]
[Out]
ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] + Sqrt[a + b*Sech[x]^2]/b
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Rubi [A] time = 0.0929296, antiderivative size = 42, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used =
{4139, 446, 80, 63, 208} \[ \frac{\sqrt{a+b \text{sech}^2(x)}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^3/Sqrt[a + b*Sech[x]^2],x]
[Out]
ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] + Sqrt[a + b*Sech[x]^2]/b
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])
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 80
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]
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 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]
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{-1+x^2}{x \sqrt{a+b x^2}} \, dx,x,\text{sech}(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x}{x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )\\ &=\frac{\sqrt{a+b \text{sech}^2(x)}}{b}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )\\ &=\frac{\sqrt{a+b \text{sech}^2(x)}}{b}-\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{\sqrt{a+b \text{sech}^2(x)}}{b}\\ \end{align*}
Mathematica [B] time = 0.211837, size = 105, normalized size = 2.5 \[ \frac{\text{sech}^2(x) (a \cosh (2 x)+a+2 b)}{2 b \sqrt{a+b \text{sech}^2(x)}}+\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \log \left (\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right )}{\sqrt{2} \sqrt{a} \sqrt{a+b \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^3/Sqrt[a + b*Sech[x]^2],x]
[Out]
(Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]*Sech[x])/(Sqrt[2]*Sqrt
[a]*Sqrt[a + b*Sech[x]^2]) + ((a + 2*b + a*Cosh[2*x])*Sech[x]^2)/(2*b*Sqrt[a + b*Sech[x]^2])
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Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( x \right ) \right ) ^{3}{\frac{1}{\sqrt{a+b \left ({\rm sech} \left (x\right ) \right ) ^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^3/(a+b*sech(x)^2)^(1/2),x)
[Out]
int(tanh(x)^3/(a+b*sech(x)^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{3}}{\sqrt{b \operatorname{sech}\left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^3/(a+b*sech(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^3/sqrt(b*sech(x)^2 + a), x)
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Fricas [B] time = 2.61953, size = 4733, normalized size = 112.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^3/(a+b*sech(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/4*((b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + b)*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*(1
4*(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)*cosh(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*cos
h(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))*sinh(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 + sin
h(x)^6)) + (b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + b)*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*sinh(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)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*sqrt(2)*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*b*cosh(x)^2 + 2*a*b*cos
h(x)*sinh(x) + a*b*sinh(x)^2 + a*b), -1/2*((b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + b)*sqrt(-a)*arct
an(sqrt(2)*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(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)*sinh(x) + sinh(x)^2))/((a^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*c
osh(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))) + (b*cosh(x)^2 + 2*b*cosh
(x)*sinh(x) + b*sinh(x)^2 + b)*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*c
osh(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)) - 2*sqrt(2)*a*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*c
osh(x)*sinh(x) + sinh(x)^2)))/(a*b*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 + a*b)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{3}{\left (x \right )}}{\sqrt{a + b \operatorname{sech}^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**3/(a+b*sech(x)**2)**(1/2),x)
[Out]
Integral(tanh(x)**3/sqrt(a + b*sech(x)**2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{3}}{\sqrt{b \operatorname{sech}\left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^3/(a+b*sech(x)^2)^(1/2),x, algorithm="giac")
[Out]
integrate(tanh(x)^3/sqrt(b*sech(x)^2 + a), x)