3.1.47 \(\int \frac {\tanh ^2(x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [47]

Optimal. Leaf size=131 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2} \]

[Out]

arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(5/2)+1/3*b*tanh(x)/a/(a+b)/(a+b*coth(x)^2)^(3/2)+1/3
*b*(7*a+4*b)*tanh(x)/a^2/(a+b)^2/(a+b*coth(x)^2)^(1/2)-1/3*(3*a+2*b)*(a+4*b)*(a+b*coth(x)^2)^(1/2)*tanh(x)/a^3
/(a+b)^2

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Rubi [A]
time = 0.17, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 483, 593, 597, 12, 385, 212} \begin {gather*} -\frac {(3 a+2 b) (a+4 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{3 a^3 (a+b)^2}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^2/(a + b*Coth[x]^2)^(5/2),x]

[Out]

ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Coth[x]^2
)^(3/2)) + (b*(7*a + 4*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Coth[x]^2]) - ((3*a + 2*b)*(a + 4*b)*Sqrt[a + b
*Coth[x]^2]*Tanh[x])/(3*a^3*(a + b)^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 483

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

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

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

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 a-4 b+4 b x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{3 a (a+b)}\\ &=\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {(3 a+2 b) (a+4 b)-2 b (7 a+4 b) x^2}{x^2 \left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a^2 (a+b)^2}\\ &=\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}-\frac {\text {Subst}\left (\int -\frac {3 a^3}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a^3 (a+b)^2}\\ &=\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{(a+b)^2}\\ &=\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 7.64, size = 1350, normalized size = 10.31 \begin {gather*} \frac {\sinh ^2(x) \left (\frac {16 b^3 \left (-i \coth (x)+i \coth ^3(x)\right )^2}{a (a+b)^2}+\frac {40 b \text {csch}^2(x)}{a+b}+\frac {160 b^2 \coth ^2(x) \text {csch}^2(x)}{3 a (a+b)}+\frac {64 b^3 \coth ^4(x) \text {csch}^2(x)}{3 a^2 (a+b)}-\frac {40 b^2 \text {csch}^4(x)}{(a+b)^2}+\frac {92 (a+b) \cosh ^2(x) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {124 b (a+b) \cosh ^2(x) \coth ^2(x) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {152 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {176 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {24 (a+b) \cosh ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{7 a^2}+\frac {88 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {32 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^4}+\frac {16 (a+b) \cosh ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {16 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {16 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {20 a \text {sech}^2(x)}{3 (a+b)}-\frac {30 a b \text {csch}^2(x) \text {sech}^2(x)}{(a+b)^2}-\frac {5 a^2 \text {sech}^4(x)}{(a+b)^2}+\frac {5 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {30 b \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {40 b^2 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {16 b^3 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {5 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {30 b \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {40 b^2 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {16 b^3 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {60 b \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {csch}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {80 b^2 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x) \text {csch}^2(x)}{a (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {32 b^3 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x) \text {csch}^2(x)}{a^2 (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {10 a \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {sech}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right ) \tanh (x)}{a^2 \sqrt {a+b \coth ^2(x)} \left (1+\frac {b \coth ^2(x)}{a}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[x]^2/(a + b*Coth[x]^2)^(5/2),x]

[Out]

(Sinh[x]^2*((16*b^3*((-I)*Coth[x] + I*Coth[x]^3)^2)/(a*(a + b)^2) + (40*b*Csch[x]^2)/(a + b) + (160*b^2*Coth[x
]^2*Csch[x]^2)/(3*a*(a + b)) + (64*b^3*Coth[x]^4*Csch[x]^2)/(3*a^2*(a + b)) - (40*b^2*Csch[x]^4)/(a + b)^2 + (
92*(a + b)*Cosh[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(105*a) + (124*b*(a + b)*Cosh[x]^2*C
oth[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(35*a^2) + (152*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*
Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (176*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*Hypergeom
etric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(105*a^4) + (24*(a + b)*Cosh[x]^2*HypergeometricPFQ[{2, 2, 2}, {1,
9/2}, ((a + b)*Cosh[x]^2)/a])/(35*a) + (16*b*(a + b)*Cosh[x]^2*Coth[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2
}, ((a + b)*Cosh[x]^2)/a])/(7*a^2) + (88*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}
, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (32*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}
, ((a + b)*Cosh[x]^2)/a])/(35*a^4) + (16*(a + b)*Cosh[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a +
b)*Cosh[x]^2)/a])/(105*a) + (16*b*(a + b)*Cosh[x]^2*Coth[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a
+ b)*Cosh[x]^2)/a])/(35*a^2) + (16*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2
}, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (16*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1,
1, 9/2}, ((a + b)*Cosh[x]^2)/a])/(105*a^4) + (20*a*Sech[x]^2)/(3*(a + b)) - (30*a*b*Csch[x]^2*Sech[x]^2)/(a +
b)^2 - (5*a^2*Sech[x]^4)/(a + b)^2 + (5*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]])/((((a + b)*Cosh[x]^2)/a)^(5/2)*Sq
rt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (30*b*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2)/(a*(((a + b)*Co
sh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (40*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth
[x]^4)/(a^2*(((a + b)*Cosh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (16*b^3*ArcSin[Sqrt[((a
+ b)*Cosh[x]^2)/a]]*Coth[x]^6)/(a^3*(((a + b)*Cosh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) +
(5*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]])/Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)] + (30*b*A
rcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2)/(a*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])
+ (40*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^4)/(a^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sin
h[x]^2)/a^2)]) + (16*b^3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^6)/(a^3*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*
Coth[x]^2)*Sinh[x]^2)/a^2)]) - (60*b*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Csch[x]^2)/((a + b)*Sqrt[-(((a + b)*C
osh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (80*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2*Csch[x]^2
)/(a*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (32*b^3*ArcSin[Sqrt[((a + b)*Cosh
[x]^2)/a]]*Coth[x]^4*Csch[x]^2)/(a^2*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (
10*a*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Sech[x]^2)/((a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[
x]^2)/a^2)]))*Tanh[x])/(a^2*Sqrt[a + b*Coth[x]^2]*(1 + (b*Coth[x]^2)/a))

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Maple [F]
time = 2.70, size = 0, normalized size = 0.00 \[\int \frac {\tanh ^{2}\left (x \right )}{\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(a+b*coth(x)^2)^(5/2),x)

[Out]

int(tanh(x)^2/(a+b*coth(x)^2)^(5/2),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)^2/(a+b*coth(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)^2/(b*coth(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. 5085 vs. \(2 (113) = 226\).
time = 1.19, size = 10729, normalized size = 81.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^10 + 10*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)*sinh(x)^9 + (a^5 + 2*a^4
*b + a^3*b^2)*sinh(x)^10 - (3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^8 - (3*a^5 - 2*a^4*b - 5*a^3*b^2 - 45*(a^5 +
2*a^4*b + a^3*b^2)*cosh(x)^2)*sinh(x)^8 + 8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^3 - (3*a^5 - 2*a^4*b - 5*a^3
*b^2)*cosh(x))*sinh(x)^7 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^6 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2 + 105*(a^5 +
2*a^4*b + a^3*b^2)*cosh(x)^4 - 14*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 2*a^4*b +
a^3*b^2)*cosh(x)^5 - 14*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^3 + 3*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x))*sinh
(x)^5 + a^5 + 2*a^4*b + a^3*b^2 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^4 + 2*(105*(a^5 + 2*a^4*b + a^3*b^2)*c
osh(x)^6 + a^5 - 2*a^4*b + 5*a^3*b^2 - 35*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^4 + 15*(a^5 - 2*a^4*b + 5*a^3*
b^2)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^7 - 7*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x
)^5 + 5*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^3 + (a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x))*sinh(x)^3 - (3*a^5 - 2*a^
4*b - 5*a^3*b^2)*cosh(x)^2 + (45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^8 - 28*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x
)^6 - 3*a^5 + 2*a^4*b + 5*a^3*b^2 + 30*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^4 + 12*(a^5 - 2*a^4*b + 5*a^3*b^2)*
cosh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^9 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^7 +
6*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^3 - (3*a^5 - 2*a^4*b - 5*a^3*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*(1
4*(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)*cosh(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)/(cos
h(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^5 + 2*a^4*b + a^3*b^2)*cosh(x)^10 + 10*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)*sinh(x)^9 + (a^5 + 2*a^4*
b + a^3*b^2)*sinh(x)^10 - (3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^8 - (3*a^5 - 2*a^4*b - 5*a^3*b^2 - 45*(a^5 + 2
*a^4*b + a^3*b^2)*cosh(x)^2)*sinh(x)^8 + 8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^3 - (3*a^5 - 2*a^4*b - 5*a^3*
b^2)*cosh(x))*sinh(x)^7 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^6 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2 + 105*(a^5 +
2*a^4*b + a^3*b^2)*cosh(x)^4 - 14*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 2*a^4*b +
a^3*b^2)*cosh(x)^5 - 14*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^3 + 3*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x))*sinh(
x)^5 + a^5 + 2*a^4*b + a^3*b^2 + 2*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^4 + 2*(105*(a^5 + 2*a^4*b + a^3*b^2)*co
sh(x)^6 + a^5 - 2*a^4*b + 5*a^3*b^2 - 35*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^4 + 15*(a^5 - 2*a^4*b + 5*a^3*b
^2)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^7 - 7*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)
^5 + 5*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^3 + (a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x))*sinh(x)^3 - (3*a^5 - 2*a^4
*b - 5*a^3*b^2)*cosh(x)^2 + (45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^8 - 28*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)
^6 - 3*a^5 + 2*a^4*b + 5*a^3*b^2 + 30*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^4 + 12*(a^5 - 2*a^4*b + 5*a^3*b^2)*c
osh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^9 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*cosh(x)^7 + 6
*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*cosh(x)^3 - (3*a^5 - 2*a^4*b - 5*a^3*b^
2)*cosh(x))*sinh(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)*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*((a + b)*cosh(x)^3 - a*cosh(x))*sinh(x) + ...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(a + b*coth(x)^2)^(5/2),x)

[Out]

int(tanh(x)^2/(a + b*coth(x)^2)^(5/2), x)

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