3.3.52 \(\int \frac {1}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [252]

Optimal. Leaf size=93 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]

[Out]

arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+b)^(5/2)+1/3*b*(5*a+2*b)*tanh(x)/a^2/(a+b)^2/(a+b*tanh(x
)^2)^(1/2)+1/3*b*tanh(x)/a/(a+b)/(a+b*tanh(x)^2)^(3/2)

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Rubi [A]
time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 425, 541, 12, 385, 212} \begin {gather*} \frac {b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tanh[x]^2)^(-5/2),x]

[Out]

ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Tanh[x]^2
)^(3/2)) + (b*(5*a + 2*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3742

Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 6.22, size = 943, normalized size = 10.14 \begin {gather*} \frac {\cosh (x) \sinh (x) \left (1575 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right )+\frac {3150 (a+b) \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x)}{a}+\frac {1575 (a+b)^2 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x)}{a^2}+2100 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}}+96 \, _2F_1\left (2,2;\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}}+24 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}}+\frac {2100 b \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^2(x)}{a}+\frac {4200 b (a+b) \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x) \tanh ^2(x)}{a^2}+\frac {2100 b (a+b)^2 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x) \tanh ^2(x)}{a^3}+\frac {2800 b \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^2(x)}{a}+\frac {168 b \, _2F_1\left (2,2;\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^2(x)}{a}+\frac {48 b \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^2(x)}{a}+\frac {840 b^2 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^4(x)}{a^2}+\frac {1680 b^2 (a+b) \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x) \tanh ^4(x)}{a^3}+\frac {840 b^2 (a+b)^2 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x) \tanh ^4(x)}{a^4}+\frac {1120 b^2 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^4(x)}{a^2}+\frac {72 b^2 \, _2F_1\left (2,2;\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^4(x)}{a^2}+\frac {24 b^2 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \tanh ^4(x)}{a^2}-1575 \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}-\frac {2100 b \tanh ^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}{a}-\frac {840 b^2 \tanh ^4(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}{a^2}\right )}{315 a \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}} \left (a+b \tanh ^2(x)\right )^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Tanh[x]^2)^(-5/2),x]

[Out]

(Cosh[x]*Sinh[x]*(1575*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]] + (3150*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2
)/a)]]*Sinh[x]^2)/a + (1575*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4)/a^2 + 2100*(-(((a + b)
*Sinh[x]^2)/a))^(3/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a] + 96*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2
)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a] + 24*HypergeometricPFQ[{2, 2, 2}, {1,
9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a] + (2100*b*
ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^2)/a + (4200*b*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*S
inh[x]^2*Tanh[x]^2)/a^2 + (2100*b*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^2)/a^3 +
(2800*b*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*Tanh[x]^2)/a + (168*b*Hypergeometri
c2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*T
anh[x]^2)/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a)
)^(7/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*Tanh[x]^2)/a + (840*b^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[
x]^4)/a^2 + (1680*b^2*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^2*Tanh[x]^4)/a^3 + (840*b^2*(a +
b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^4)/a^4 + (1120*b^2*(-(((a + b)*Sinh[x]^2)/a))^(3
/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*Tanh[x]^4)/a^2 + (72*b^2*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]
^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*Tanh[x]^4)/a^2 + (24*b^2*Hypergeome
tricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[Cosh[x]^2 + (b*Si
nh[x]^2)/a]*Tanh[x]^4)/a^2 - 1575*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)] - (2100*b*Tanh[
x]^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a - (840*b^2*Tanh[x]^4*Sqrt[-(((a + b)*Cosh
[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a^2))/(315*a*(-(((a + b)*Sinh[x]^2)/a))^(5/2)*Sqrt[Cosh[x]^2 + (b*Si
nh[x]^2)/a]*(a + b*Tanh[x]^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(549\) vs. \(2(79)=158\).
time = 0.72, size = 550, normalized size = 5.91 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/(a+b)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(tanh(x)-1)+2*b)/(4*b*(a+b)-4*b^2
)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(tanh(x)-1)+2*b)/(b*(tanh(x)-1)^
2+2*b*(tanh(x)-1)+a+b)^(1/2))-1/2/(a+b)*(1/(a+b)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)-2*b/(a+b)*(2*b*(t
anh(x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)-1/(a+b)^(3/2)*ln((2*a+2*b+2*b*(ta
nh(x)-1)+2*(a+b)^(1/2)*(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2))/(tanh(x)-1)))+1/6/(a+b)/(b*(1+tanh(x))^2-2
*b*(1+tanh(x))+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(1+tanh(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*(1+tanh(x))^2-2*b*(1+tan
h(x))+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(1+tanh(x))-2*b)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))
+1/2/(a+b)*(1/(a+b)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)+2*b/(a+b)*(2*b*(1+tanh(x))-2*b)/(4*b*(a+b)-4*b
^2)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)-1/(a+b)^(3/2)*ln((2*a+2*b-2*b*(1+tanh(x))+2*(a+b)^(1/2)*(b*(1+
tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))/(1+tanh(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(1/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*tanh(x)^2 + a)^(-5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3152 vs. \(2 (79) = 158\).
time = 0.72, size = 6933, normalized size = 74.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^3*b
+ a^2*b^2)*sinh(x)^8 + 4*(a^4 - a^2*b^2)*cosh(x)^6 + 4*(a^4 - a^2*b^2 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2
)*sinh(x)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^4 - a^2*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^4 - 2*a
^3*b + 3*a^2*b^2)*cosh(x)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^4 + 3*a^4 - 2*a^3*b + 3*a^2*b^2 + 30*(a^
4 - a^2*b^2)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^5 + 10*(a
^4 - a^2*b^2)*cosh(x)^3 + (3*a^4 - 2*a^3*b + 3*a^2*b^2)*cosh(x))*sinh(x)^3 + 4*(a^4 - a^2*b^2)*cosh(x)^2 + 4*(
7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^6 + 15*(a^4 - a^2*b^2)*cosh(x)^4 + a^4 - a^2*b^2 + 3*(3*a^4 - 2*a^3*b + 3*
a^2*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^7 + 3*(a^4 - a^2*b^2)*cosh(x)^5 + (3*a^4
- 2*a^3*b + 3*a^2*b^2)*cosh(x)^3 + (a^4 - a^2*b^2)*cosh(x))*sinh(x))*sqrt(a + b)*log(-((a*b^2 + b^3)*cosh(x)^8
+ 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 - 2*(a*b^2 + 2*b^3)*cosh(x)^6 - 2*(a*b^2 + 2*b^
3 - 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 - 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)
^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 -
30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 - 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3
- a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 2*(a^3 - 3*a*b^2 - 2*b^3)*cos
h(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 - 15*(a*b^2 + 2*b^3)*cosh(x)^4 + a^3 - 3*a*b^2 - 2*b^3 + 3*(a^3 - a^2*b
+ 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(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 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^3 - (a^2 -
2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^2)*sinh(x)^2 - a^2 - 2*a*
b - b^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*sinh(x))*sqrt(a + b)*sqrt(((a
+ b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*(a*b^2 + b^3)*
cosh(x)^7 - 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^3 + (a^3 - 3*a*b^2 - 2*b^3)*
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 + sinh(x)^6)) + 3*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*
b + a^2*b^2)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(x)^8 + 4*(a^4 - a^2*b^2)*cosh(x)^6 + 4*(a^4 -
a^2*b^2 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^4
- a^2*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cosh(x)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*c
osh(x)^4 + 3*a^4 - 2*a^3*b + 3*a^2*b^2 + 30*(a^4 - a^2*b^2)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8
*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^5 + 10*(a^4 - a^2*b^2)*cosh(x)^3 + (3*a^4 - 2*a^3*b + 3*a^2*b^2)*cosh(x)
)*sinh(x)^3 + 4*(a^4 - a^2*b^2)*cosh(x)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^6 + 15*(a^4 - a^2*b^2)*cosh
(x)^4 + a^4 - a^2*b^2 + 3*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*co
sh(x)^7 + 3*(a^4 - a^2*b^2)*cosh(x)^5 + (3*a^4 - 2*a^3*b + 3*a^2*b^2)*cosh(x)^3 + (a^4 - a^2*b^2)*cosh(x))*sin
h(x))*sqrt(a + b)*log(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*a*cosh(x)^2 + 2
*(3*(a + b)*cosh(x)^2 + a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a + b)*sqr
t(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + b)*co
sh(x)^3 + a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 8*sqrt(2)*((3*a^3*b + 7*a
^2*b^2 + 5*a*b^3 + b^4)*cosh(x)^6 + 6*(3*a^3*b + 7*a^2*b^2 + 5*a*b^3 + b^4)*cosh(x)*sinh(x)^5 + (3*a^3*b + 7*a
^2*b^2 + 5*a*b^3 + b^4)*sinh(x)^6 + 3*(a^3*b - a^2*b^2 - 3*a*b^3 - b^4)*cosh(x)^4 + 3*(a^3*b - a^2*b^2 - 3*a*b
^3 - b^4 + 5*(3*a^3*b + 7*a^2*b^2 + 5*a*b^3 + b^4)*cosh(x)^2)*sinh(x)^4 - 3*a^3*b - 7*a^2*b^2 - 5*a*b^3 - b^4
+ 4*(5*(3*a^3*b + 7*a^2*b^2 + 5*a*b^3 + b^4)*cosh(x)^3 + 3*(a^3*b - a^2*b^2 - 3*a*b^3 - b^4)*cosh(x))*sinh(x)^
3 - 3*(a^3*b - a^2*b^2 - 3*a*b^3 - b^4)*cosh(x)^2 + 3*(5*(3*a^3*b + 7*a^2*b^2 + 5*a*b^3 + b^4)*cosh(x)^4 - a^3
*b + a^2*b^2 + 3*a*b^3 + b^4 + 6*(a^3*b - a^2*b^2 - 3*a*b^3 - b^4)*cosh(x)^2)*sinh(x)^2 + 6*((3*a^3*b + 7*a^2*
b^2 + 5*a*b^3 + b^4)*cosh(x)^5 + 2*(a^3*b - a^2*b^2 - 3*a*b^3 - b^4)*cosh(x)^3 - (a^3*b - a^2*b^2 - 3*a*b^3 -
b^4)*cosh(x))*sinh(x))*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + s
inh(x)^2)))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cosh(x)^8 + 8*(a^7 + 5*a^6*b + 10
*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cosh(x)*sinh(x)^7 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*
a^3*b^4 + a^2*b^5)*sinh(x)^8 + a^7 + 5*a^6*b + ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tanh(x)**2)**(-5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (79) = 158\).
time = 0.62, size = 738, normalized size = 7.94 \begin {gather*} -\frac {\sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {2 \, {\left ({\left ({\left (\frac {{\left (3 \, a^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}\right )} e^{\left (2 \, x\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}} + \frac {3 \, {\left (a^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, {\left (a^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, a^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )}}{3 \, {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(a + b)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a +
b))*(a + b) - sqrt(a + b)*(a - b)))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(
2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b)))/(a^3 + 3*a^2*b + 3*a*b^
2 + b^3) + 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*
x) + a + b) - sqrt(a + b)))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 2/3*((((3*a^6*b^3 + 16*a^5*b^4 + 35*a^4*b^5 + 40
*a^3*b^6 + 25*a^2*b^7 + 8*a*b^8 + b^9)*e^(2*x)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6
*a^3*b^7 + a^2*b^8) + 3*(a^6*b^3 + 2*a^5*b^4 - 3*a^4*b^5 - 12*a^3*b^6 - 13*a^2*b^7 - 6*a*b^8 - b^9)/(a^8*b^2 +
6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) - 3*(a^6*b^3 + 2*a^5*b^4 - 3
*a^4*b^5 - 12*a^3*b^6 - 13*a^2*b^7 - 6*a*b^8 - b^9)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^
6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) - (3*a^6*b^3 + 16*a^5*b^4 + 35*a^4*b^5 + 40*a^3*b^6 + 25*a^2*b^7 + 8*a*b^8 +
b^9)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))/(a*e^(4*x) + b*e^(4*
x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tanh(x)^2)^(5/2),x)

[Out]

int(1/(a + b*tanh(x)^2)^(5/2), x)

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