3.3.3 \(\int \frac {\tanh ^5(x)}{(a+b \text {sech}^2(x))^{3/2}} \, dx\) [203]
Optimal. Leaf size=68 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {(a+b)^2}{a b^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\sqrt {a+b \text {sech}^2(x)}}{b^2} \]
[Out]
arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))/a^(3/2)-(a+b)^2/a/b^2/(a+b*sech(x)^2)^(1/2)-(a+b*sech(x)^2)^(1/2)/b^2
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Rubi [A]
time = 0.10, antiderivative size = 68, 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 = {4224, 457, 89,
65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {(a+b)^2}{a b^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\sqrt {a+b \text {sech}^2(x)}}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^5/(a + b*Sech[x]^2)^(3/2),x]
[Out]
ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(3/2) - (a + b)^2/(a*b^2*Sqrt[a + b*Sech[x]^2]) - Sqrt[a + b*Sech[x]^
2]/b^2
Rule 65
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 89
Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 457
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 4224
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)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx &=-\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x \left (a+b x^2\right )^{3/2}} \, dx,x,\text {sech}(x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^2}{x (a+b x)^{3/2}} \, dx,x,\text {sech}^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \left (-\frac {(a+b)^2}{a b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}+\frac {1}{a x \sqrt {a+b x}}\right ) \, dx,x,\text {sech}^2(x)\right )\right )\\ &=-\frac {(a+b)^2}{a b^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\sqrt {a+b \text {sech}^2(x)}}{b^2}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )}{2 a}\\ &=-\frac {(a+b)^2}{a b^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\sqrt {a+b \text {sech}^2(x)}}{b^2}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {sech}^2(x)}\right )}{a b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {(a+b)^2}{a b^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\sqrt {a+b \text {sech}^2(x)}}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 129, normalized size = 1.90 \begin {gather*} \frac {\text {sech}^3(x) \left (\frac {\sqrt {2} (a+2 b+a \cosh (2 x))^{3/2} \log \left (\sqrt {2} \sqrt {a} \cosh (x)+\sqrt {a+2 b+a \cosh (2 x)}\right )}{a^{3/2}}-\frac {(a+2 b+a \cosh (2 x)) \left (2 a^2+4 a b+b^2+\left (2 a^2+2 a b+b^2\right ) \cosh (2 x)\right ) \text {sech}(x)}{a b^2}\right )}{4 \left (a+b \text {sech}^2(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^5/(a + b*Sech[x]^2)^(3/2),x]
[Out]
(Sech[x]^3*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])
/a^(3/2) - ((a + 2*b + a*Cosh[2*x])*(2*a^2 + 4*a*b + b^2 + (2*a^2 + 2*a*b + b^2)*Cosh[2*x])*Sech[x])/(a*b^2)))
/(4*(a + b*Sech[x]^2)^(3/2))
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Maple [F]
time = 1.68, size = 0, normalized size = 0.00 \[\int \frac {\tanh ^{5}\left (x \right )}{\left (a +b \mathrm {sech}\left (x \right )^{2}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^5/(a+b*sech(x)^2)^(3/2),x)
[Out]
int(tanh(x)^5/(a+b*sech(x)^2)^(3/2),x)
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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*sech(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^5/(b*sech(x)^2 + a)^(3/2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1312 vs.
\(2 (58) = 116\).
time = 0.54, size = 3360, normalized size = 49.41 \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*sech(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*((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 + 4*b^3)*cosh(x)^4 + (15*a*b^2
*cosh(x)^2 + 3*a*b^2 + 4*b^3)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 + (3*a*b^2 + 4*b^3)*cosh(x))*sinh(x)^3 + a*b^2
+ (3*a*b^2 + 4*b^3)*cosh(x)^2 + (15*a*b^2*cosh(x)^4 + 3*a*b^2 + 4*b^3 + 6*(3*a*b^2 + 4*b^3)*cosh(x)^2)*sinh(x)
^2 + 2*(3*a*b^2*cosh(x)^5 + 2*(3*a*b^2 + 4*b^3)*cosh(x)^3 + (3*a*b^2 + 4*b^3)*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
)*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)*cos
h(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*sin
h(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)
)*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*s
inh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + (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 + 4*b^3)*cosh(x)^4 + (15*a*b^2*cosh(x)^2 + 3*a*b^2 + 4*b^3)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 + (3*a
*b^2 + 4*b^3)*cosh(x))*sinh(x)^3 + a*b^2 + (3*a*b^2 + 4*b^3)*cosh(x)^2 + (15*a*b^2*cosh(x)^4 + 3*a*b^2 + 4*b^3
+ 6*(3*a*b^2 + 4*b^3)*cosh(x)^2)*sinh(x)^2 + 2*(3*a*b^2*cosh(x)^5 + 2*(3*a*b^2 + 4*b^3)*cosh(x)^3 + (3*a*b^2
+ 4*b^3)*cosh(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*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)*((2*a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 4*(2*a
^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^3 + (2*a^3 + 2*a^2*b + a*b^2)*sinh(x)^4 + 2*a^3 + 2*a^2*b + a*b^2 + 2*(2
*a^3 + 4*a^2*b + a*b^2)*cosh(x)^2 + 2*(2*a^3 + 4*a^2*b + a*b^2 + 3*(2*a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x
)^2 + 4*((2*a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + (2*a^3 + 4*a^2*b + a*b^2)*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^3*b^2*cosh(x)^6 + 6*a^3*b^2*cosh(x)*s
inh(x)^5 + a^3*b^2*sinh(x)^6 + a^3*b^2 + (3*a^3*b^2 + 4*a^2*b^3)*cosh(x)^4 + (15*a^3*b^2*cosh(x)^2 + 3*a^3*b^2
+ 4*a^2*b^3)*sinh(x)^4 + 4*(5*a^3*b^2*cosh(x)^3 + (3*a^3*b^2 + 4*a^2*b^3)*cosh(x))*sinh(x)^3 + (3*a^3*b^2 + 4
*a^2*b^3)*cosh(x)^2 + (15*a^3*b^2*cosh(x)^4 + 3*a^3*b^2 + 4*a^2*b^3 + 6*(3*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2)*sin
h(x)^2 + 2*(3*a^3*b^2*cosh(x)^5 + 2*(3*a^3*b^2 + 4*a^2*b^3)*cosh(x)^3 + (3*a^3*b^2 + 4*a^2*b^3)*cosh(x))*sinh(
x)), -1/2*((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 + 4*b^3)*cosh(x)^4 + (15*
a*b^2*cosh(x)^2 + 3*a*b^2 + 4*b^3)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 + (3*a*b^2 + 4*b^3)*cosh(x))*sinh(x)^3 + a
*b^2 + (3*a*b^2 + 4*b^3)*cosh(x)^2 + (15*a*b^2*cosh(x)^4 + 3*a*b^2 + 4*b^3 + 6*(3*a*b^2 + 4*b^3)*cosh(x)^2)*si
nh(x)^2 + 2*(3*a*b^2*cosh(x)^5 + 2*(3*a*b^2 + 4*b^3)*cosh(x)^3 + (3*a*b^2 + 4*b^3)*cosh(x))*sinh(x))*sqrt(-a)*
arctan(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)*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))) + (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 + 4*b^3)*cosh(x)^4 + (15*a*b^2*cosh(x)^2 + 3*a*b^2 + 4*
b^3)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 + (3*a*b^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**5/(a+b*sech(x)**2)**(3/2),x)
[Out]
Integral(tanh(x)**5/(a + b*sech(x)**2)**(3/2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^5/(a+b*sech(x)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type
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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}{{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^5/(a + b/cosh(x)^2)^(3/2),x)
[Out]
int(tanh(x)^5/(a + b/cosh(x)^2)^(3/2), x)
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