3.236 \(\int \frac{\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=224 \[ -\frac{2 a^5 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{8 a^3 b^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{\left (-3 a^2 b^2+2 a^4-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}+\frac{\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))} \]

[Out]

(-2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (8*a^3*b^2*ArcTanh[(b - a*Tanh[x/2])/S
qrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (4*a^3*b*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (
a^4*b*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2 - ((2*a^4 - 3*a^2*b^2 - b
^4)*Tanh[x])/(a^2 + b^2)^3 - ((a^2 - b^2)*Tanh[x]^3)/(3*(a^2 + b^2)^2)

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Rubi [A]  time = 0.434703, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {2731, 2664, 12, 2660, 618, 206, 2669, 3767, 8} \[ -\frac{2 a^5 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{8 a^3 b^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{\left (-3 a^2 b^2+2 a^4-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}+\frac{\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^4/(a + b*Sinh[x])^2,x]

[Out]

(-2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (8*a^3*b^2*ArcTanh[(b - a*Tanh[x/2])/S
qrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (4*a^3*b*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (
a^4*b*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2 - ((2*a^4 - 3*a^2*b^2 - b
^4)*Tanh[x])/(a^2 + b^2)^3 - ((a^2 - b^2)*Tanh[x]^3)/(3*(a^2 + b^2)^2)

Rule 2731

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Int[ExpandIntegrand
[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^m)/(1 - Sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a
^2 - b^2, 0] && IntegersQ[m, p/2]

Rule 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx &=\int \left (\frac{a^4}{\left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac{4 a^3 b^2}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{\text{sech}^4(x) \left (a^2 \left (1-\frac{b^2}{a^2}\right )-2 a b \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac{\text{sech}^2(x) \left (-2 a^4 \left (1-\frac{3 a^2 b^2+b^4}{2 a^4}\right )+4 a^3 b \sinh (x)\right )}{\left (a^2+b^2\right )^3}\right ) \, dx\\ &=\frac{\int \text{sech}^2(x) \left (-2 a^4 \left (1-\frac{3 a^2 b^2+b^4}{2 a^4}\right )+4 a^3 b \sinh (x)\right ) \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (4 a^3 b^2\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac{\int \text{sech}^4(x) \left (a^2 \left (1-\frac{b^2}{a^2}\right )-2 a b \sinh (x)\right ) \, dx}{\left (a^2+b^2\right )^2}+\frac{a^4 \int \frac{1}{(a+b \sinh (x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{a^4 \int \frac{a}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (8 a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}+\frac{\left (a^2-b^2\right ) \int \text{sech}^4(x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^4-3 a^2 b^2-b^4\right ) \int \text{sech}^2(x) \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{a^5 \int \frac{1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (16 a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}+\frac{\left (i \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )}{\left (a^2+b^2\right )^2}-\frac{\left (i \left (2 a^4-3 a^2 b^2-b^4\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))}{\left (a^2+b^2\right )^3}\\ &=\frac{8 a^3 b^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac{\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac{\left (2 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=\frac{8 a^3 b^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac{\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{\left (4 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=-\frac{2 a^5 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{8 a^3 b^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{4 a^3 b \text{sech}(x)}{\left (a^2+b^2\right )^3}+\frac{2 a b \text{sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac{\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac{\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.389672, size = 144, normalized size = 0.64 \[ \frac{\left (9 a^2 b^2-4 a^4+b^4\right ) \tanh (x)+\frac{6 a^3 \left (a^2-4 b^2\right ) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\left (a^2+b^2\right ) \text{sech}^3(x) \left (\left (a^2-b^2\right ) \sinh (x)+2 a b\right )-12 a^3 b \text{sech}(x)-\frac{3 a^4 b \cosh (x)}{a+b \sinh (x)}}{3 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^4/(a + b*Sinh[x])^2,x]

[Out]

((6*a^3*(a^2 - 4*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 12*a^3*b*Sech[x] - (3*a^4
*b*Cosh[x])/(a + b*Sinh[x]) + (a^2 + b^2)*Sech[x]^3*(2*a*b + (a^2 - b^2)*Sinh[x]) + (-4*a^4 + 9*a^2*b^2 + b^4)
*Tanh[x])/(3*(a^2 + b^2)^3)

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Maple [A]  time = 0.062, size = 262, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( -{a}^{4}+3\,{a}^{2}{b}^{2} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+ \left ( -2\,{a}^{3}b+2\,a{b}^{3} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+ \left ( -10/3\,{a}^{4}+6\,{a}^{2}{b}^{2}+4/3\,{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-8\,{a}^{3}b \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+ \left ( -{a}^{4}+3\,{a}^{2}{b}^{2} \right ) \tanh \left ( x/2 \right ) -10/3\,{a}^{3}b+2/3\,a{b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,{\frac{{a}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) } \left ({\frac{-{b}^{2}\tanh \left ( x/2 \right ) -ab}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a}}-{\frac{{a}^{2}-4\,{b}^{2}}{\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^4/(a+b*sinh(x))^2,x)

[Out]

2/(a^2+b^2)/(a^4+2*a^2*b^2+b^4)*((-a^4+3*a^2*b^2)*tanh(1/2*x)^5+(-2*a^3*b+2*a*b^3)*tanh(1/2*x)^4+(-10/3*a^4+6*
a^2*b^2+4/3*b^4)*tanh(1/2*x)^3-8*a^3*b*tanh(1/2*x)^2+(-a^4+3*a^2*b^2)*tanh(1/2*x)-10/3*a^3*b+2/3*a*b^3)/(tanh(
1/2*x)^2+1)^3-2*a^3/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-b^2*tanh(1/2*x)-a*b)/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)-
(a^2-4*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.72786, size = 8182, normalized size = 36.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

-1/3*(6*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x)^7 + 6*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*sinh(x)^7 - 14*a^6*b + 4*a^4
*b^3 + 20*a^2*b^5 + 2*b^7 - 6*(7*a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x)^6 - 6*(7*a^6*b + 10*a^4*b^3 + 4
*a^2*b^5 + b^7 - 7*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x))*sinh(x)^6 + 2*(21*a^7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b
^6)*cosh(x)^5 + 2*(21*a^7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b^6 + 63*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x)^2 - 18*(
7*a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x))*sinh(x)^5 - 2*(41*a^6*b + 34*a^4*b^3 - 10*a^2*b^5 - 3*b^7)*co
sh(x)^4 - 2*(41*a^6*b + 34*a^4*b^3 - 10*a^2*b^5 - 3*b^7 - 105*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x)^3 + 45*(7*
a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x)^2 - 5*(21*a^7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b^6)*cosh(x))*sinh(x)
^4 + 2*(21*a^7 - 11*a^5*b^2 - 40*a^3*b^4 - 8*a*b^6)*cosh(x)^3 + 2*(21*a^7 - 11*a^5*b^2 - 40*a^3*b^4 - 8*a*b^6
+ 105*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x)^4 - 60*(7*a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x)^3 + 10*(21
*a^7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b^6)*cosh(x)^2 - 4*(41*a^6*b + 34*a^4*b^3 - 10*a^2*b^5 - 3*b^7)*cosh(x))*sin
h(x)^3 - 2*(35*a^6*b + 26*a^4*b^3 - 8*a^2*b^5 + b^7)*cosh(x)^2 - 2*(35*a^6*b + 26*a^4*b^3 - 8*a^2*b^5 + b^7 -
63*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*cosh(x)^5 + 45*(7*a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x)^4 - 10*(21*a^
7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b^6)*cosh(x)^3 + 6*(41*a^6*b + 34*a^4*b^3 - 10*a^2*b^5 - 3*b^7)*cosh(x)^2 - 3*(
21*a^7 - 11*a^5*b^2 - 40*a^3*b^4 - 8*a*b^6)*cosh(x))*sinh(x)^2 - 3*((a^5*b - 4*a^3*b^3)*cosh(x)^8 + (a^5*b - 4
*a^3*b^3)*sinh(x)^8 + 2*(a^6 - 4*a^4*b^2)*cosh(x)^7 + 2*(a^6 - 4*a^4*b^2 + 4*(a^5*b - 4*a^3*b^3)*cosh(x))*sinh
(x)^7 + 2*(a^5*b - 4*a^3*b^3)*cosh(x)^6 + 2*(a^5*b - 4*a^3*b^3 + 14*(a^5*b - 4*a^3*b^3)*cosh(x)^2 + 7*(a^6 - 4
*a^4*b^2)*cosh(x))*sinh(x)^6 - a^5*b + 4*a^3*b^3 + 6*(a^6 - 4*a^4*b^2)*cosh(x)^5 + 2*(3*a^6 - 12*a^4*b^2 + 28*
(a^5*b - 4*a^3*b^3)*cosh(x)^3 + 21*(a^6 - 4*a^4*b^2)*cosh(x)^2 + 6*(a^5*b - 4*a^3*b^3)*cosh(x))*sinh(x)^5 + 10
*(7*(a^5*b - 4*a^3*b^3)*cosh(x)^4 + 7*(a^6 - 4*a^4*b^2)*cosh(x)^3 + 3*(a^5*b - 4*a^3*b^3)*cosh(x)^2 + 3*(a^6 -
4*a^4*b^2)*cosh(x))*sinh(x)^4 + 6*(a^6 - 4*a^4*b^2)*cosh(x)^3 + 2*(3*a^6 - 12*a^4*b^2 + 28*(a^5*b - 4*a^3*b^3
)*cosh(x)^5 + 35*(a^6 - 4*a^4*b^2)*cosh(x)^4 + 20*(a^5*b - 4*a^3*b^3)*cosh(x)^3 + 30*(a^6 - 4*a^4*b^2)*cosh(x)
^2)*sinh(x)^3 - 2*(a^5*b - 4*a^3*b^3)*cosh(x)^2 + 2*(14*(a^5*b - 4*a^3*b^3)*cosh(x)^6 - a^5*b + 4*a^3*b^3 + 21
*(a^6 - 4*a^4*b^2)*cosh(x)^5 + 15*(a^5*b - 4*a^3*b^3)*cosh(x)^4 + 30*(a^6 - 4*a^4*b^2)*cosh(x)^3 + 9*(a^6 - 4*
a^4*b^2)*cosh(x))*sinh(x)^2 + 2*(a^6 - 4*a^4*b^2)*cosh(x) + 2*(4*(a^5*b - 4*a^3*b^3)*cosh(x)^7 + 7*(a^6 - 4*a^
4*b^2)*cosh(x)^6 + a^6 - 4*a^4*b^2 + 6*(a^5*b - 4*a^3*b^3)*cosh(x)^5 + 15*(a^6 - 4*a^4*b^2)*cosh(x)^4 + 9*(a^6
- 4*a^4*b^2)*cosh(x)^2 - 2*(a^5*b - 4*a^3*b^3)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sin
h(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x
) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) + 2*(11*a^7 + 5*a^5*b^2 - 8
*a^3*b^4 - 2*a*b^6)*cosh(x) + 2*(11*a^7 + 5*a^5*b^2 - 8*a^3*b^4 - 2*a*b^6 + 21*(a^7 - 3*a^5*b^2 - 4*a^3*b^4)*c
osh(x)^6 - 18*(7*a^6*b + 10*a^4*b^3 + 4*a^2*b^5 + b^7)*cosh(x)^5 + 5*(21*a^7 - a^5*b^2 - 20*a^3*b^4 + 2*a*b^6)
*cosh(x)^4 - 4*(41*a^6*b + 34*a^4*b^3 - 10*a^2*b^5 - 3*b^7)*cosh(x)^3 + 3*(21*a^7 - 11*a^5*b^2 - 40*a^3*b^4 -
8*a*b^6)*cosh(x)^2 - 2*(35*a^6*b + 26*a^4*b^3 - 8*a^2*b^5 + b^7)*cosh(x))*sinh(x))/(a^8*b + 4*a^6*b^3 + 6*a^4*
b^5 + 4*a^2*b^7 + b^9 - (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^8 - (a^8*b + 4*a^6*b^3 + 6*a
^4*b^5 + 4*a^2*b^7 + b^9)*sinh(x)^8 - 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^7 - 2*(a^9 +
4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + 4*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x))*sinh
(x)^7 - 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^6 - 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a
^2*b^7 + b^9 + 14*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^2 + 7*(a^9 + 4*a^7*b^2 + 6*a^5*b^4
+ 4*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^6 - 6*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^5 - 2*(
3*a^9 + 12*a^7*b^2 + 18*a^5*b^4 + 12*a^3*b^6 + 3*a*b^8 + 28*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*
cosh(x)^3 + 21*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^2 + 6*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5
+ 4*a^2*b^7 + b^9)*cosh(x))*sinh(x)^5 - 10*(7*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^4 + 7*
(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 + 3*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b
^9)*cosh(x)^2 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^4 - 6*(a^9 + 4*a^7*b^2 +
6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 - 2*(3*a^9 + 12*a^7*b^2 + 18*a^5*b^4 + 12*a^3*b^6 + 3*a*b^8 + 28*(a^8
*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^5 + 35*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)
*cosh(x)^4 + 20*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^3 + 30*(a^9 + 4*a^7*b^2 + 6*a^5*b^4
+ 4*a^3*b^6 + a*b^8)*cosh(x)^2)*sinh(x)^3 + 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^2 + 2*
(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9 - 14*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x
)^6 - 21*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^5 - 15*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a
^2*b^7 + b^9)*cosh(x)^4 - 30*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 - 9*(a^9 + 4*a^7*b^2
+ 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^2 - 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh
(x) - 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + 4*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)
*cosh(x)^7 + 7*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^6 + 6*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5
+ 4*a^2*b^7 + b^9)*cosh(x)^5 + 15*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^4 + 9*(a^9 + 4*a^7
*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^2 - 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x))
*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{4}{\left (x \right )}}{\left (a + b \sinh{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**4/(a+b*sinh(x))**2,x)

[Out]

Integral(tanh(x)**4/(a + b*sinh(x))**2, x)

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Giac [A]  time = 1.22414, size = 394, normalized size = 1.76 \begin{align*} \frac{{\left (a^{5} - 4 \, a^{3} b^{2}\right )} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (a^{5} e^{x} - a^{4} b\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac{2 \,{\left (12 \, a^{3} b e^{\left (5 \, x\right )} - 6 \, a^{4} e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 16 \, a^{3} b e^{\left (3 \, x\right )} - 8 \, a b^{3} e^{\left (3 \, x\right )} - 6 \, a^{4} e^{\left (2 \, x\right )} + 18 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 12 \, a^{3} b e^{x} - 4 \, a^{4} + 9 \, a^{2} b^{2} + b^{4}\right )}}{3 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

(a^5 - 4*a^3*b^2)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(a^5*e^x - a^4*b)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(b*e^(
2*x) + 2*a*e^x - b)) - 2/3*(12*a^3*b*e^(5*x) - 6*a^4*e^(4*x) + 9*a^2*b^2*e^(4*x) + 3*b^4*e^(4*x) + 16*a^3*b*e^
(3*x) - 8*a*b^3*e^(3*x) - 6*a^4*e^(2*x) + 18*a^2*b^2*e^(2*x) + 12*a^3*b*e^x - 4*a^4 + 9*a^2*b^2 + b^4)/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*(e^(2*x) + 1)^3)