3.18 \(\int \frac{\text{csch}^6(x)}{a+b \cosh ^2(x)} \, dx\)
Optimal. Leaf size=89 \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\coth ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \coth ^3(x)}{3 (a+b)^2} \]
[Out]
-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Coth[x])/(a +
b)^3 + ((2*a + 3*b)*Coth[x]^3)/(3*(a + b)^2) - Coth[x]^5/(5*(a + b))
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Rubi [A] time = 0.108046, antiderivative size = 89, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{3191, 390, 208} \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\coth ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \coth ^3(x)}{3 (a+b)^2} \]
Antiderivative was successfully verified.
[In]
Int[Csch[x]^6/(a + b*Cosh[x]^2),x]
[Out]
-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Coth[x])/(a +
b)^3 + ((2*a + 3*b)*Coth[x]^3)/(3*(a + b)^2) - Coth[x]^5/(5*(a + b))
Rule 3191
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 390
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]
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{\text{csch}^6(x)}{a+b \cosh ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{a-(a+b) x^2} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2+3 a b+3 b^2}{(a+b)^3}-\frac{(2 a+3 b) x^2}{(a+b)^2}+\frac{x^4}{a+b}+\frac{b^3}{(a+b)^3 \left (a-(a+b) x^2\right )}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac{\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}+\frac{(2 a+3 b) \coth ^3(x)}{3 (a+b)^2}-\frac{\coth ^5(x)}{5 (a+b)}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a-(a+b) x^2} \, dx,x,\coth (x)\right )}{(a+b)^3}\\ &=-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}+\frac{(2 a+3 b) \coth ^3(x)}{3 (a+b)^2}-\frac{\coth ^5(x)}{5 (a+b)}\\ \end{align*}
Mathematica [A] time = 0.35351, size = 92, normalized size = 1.03 \[ -\frac{\coth (x) \left (-\left (4 a^2+13 a b+9 b^2\right ) \text{csch}^2(x)+8 a^2+3 (a+b)^2 \text{csch}^4(x)+26 a b+33 b^2\right )}{15 (a+b)^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[x]^6/(a + b*Cosh[x]^2),x]
[Out]
-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - (Coth[x]*(8*a^2 + 26*a*b + 33*b^2 -
(4*a^2 + 13*a*b + 9*b^2)*Csch[x]^2 + 3*(a + b)^2*Csch[x]^4))/(15*(a + b)^3)
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Maple [B] time = 0.047, size = 310, normalized size = 3.5 \begin{align*} -{\frac{{a}^{2}}{160\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{ab}{80\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{{b}^{2}}{160\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{5\,{a}^{2}}{96\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{7\,ab}{48\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{3\,{b}^{2}}{32\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{5\,{a}^{2}}{16\, \left ( a+b \right ) ^{3}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{ab}{ \left ( a+b \right ) ^{3}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{19\,{b}^{2}}{16\, \left ( a+b \right ) ^{3}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{160\,a+160\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}}+{\frac{5\,a}{96\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{3\,b}{32\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{5\,{a}^{2}}{16\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{ab}{ \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{19\,{b}^{2}}{16\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{{b}^{3}}{2}\ln \left ( \sqrt{a+b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\sqrt{a}\tanh \left ( x/2 \right ) +\sqrt{a+b} \right ) \left ( a+b \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a}}}}+{\frac{{b}^{3}}{2}\ln \left ( -\sqrt{a+b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\sqrt{a}\tanh \left ( x/2 \right ) -\sqrt{a+b} \right ) \left ( a+b \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^6/(a+b*cosh(x)^2),x)
[Out]
-1/160/(a+b)^3*tanh(1/2*x)^5*a^2-1/80/(a+b)^3*tanh(1/2*x)^5*a*b-1/160/(a+b)^3*tanh(1/2*x)^5*b^2+5/96/(a+b)^3*t
anh(1/2*x)^3*a^2+7/48/(a+b)^3*a*tanh(1/2*x)^3*b+3/32/(a+b)^3*b^2*tanh(1/2*x)^3-5/16/(a+b)^3*a^2*tanh(1/2*x)-1/
(a+b)^3*a*b*tanh(1/2*x)-19/16/(a+b)^3*b^2*tanh(1/2*x)-1/160/(a+b)/tanh(1/2*x)^5+5/96/(a+b)^2/tanh(1/2*x)^3*a+3
/32/(a+b)^2/tanh(1/2*x)^3*b-5/16/(a+b)^3/tanh(1/2*x)*a^2-1/(a+b)^3/tanh(1/2*x)*a*b-19/16/(a+b)^3/tanh(1/2*x)*b
^2-1/2*b^3/(a+b)^(7/2)/a^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*a^(1/2)*tanh(1/2*x)+(a+b)^(1/2))+1/2*b^3/(a+b)^(
7/2)/a^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*x)^2+2*a^(1/2)*tanh(1/2*x)-(a+b)^(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(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.4546, size = 12037, normalized size = 135.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="fricas")
[Out]
[-1/30*(60*(a^2*b^2 + a*b^3)*cosh(x)^8 + 480*(a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + 60*(a^2*b^2 + a*b^3)*sinh(x
)^8 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^6 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3 - 14*(a^2*b^2 + a*b^3)*co
sh(x)^2)*sinh(x)^6 + 240*(14*(a^2*b^2 + a*b^3)*cosh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x))*sinh(x)^5
+ 40*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^4 + 40*(105*(a^2*b^2 + a*b^3)*cosh(x)^4 + 8*a^4 + 31*a
^3*b + 47*a^2*b^2 + 24*a*b^3 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^2)*sinh(x)^4 + 32*a^4 + 136*a^3*b + 23
6*a^2*b^2 + 132*a*b^3 + 160*(21*(a^2*b^2 + a*b^3)*cosh(x)^5 - 15*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^3 + (8*
a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x))*sinh(x)^3 - 40*(4*a^4 + 17*a^3*b + 28*a^2*b^2 + 15*a*b^3)*cos
h(x)^2 + 40*(42*(a^2*b^2 + a*b^3)*cosh(x)^6 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^4 - 4*a^4 - 17*a^3*b -
28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^2)*sinh(x)^2 - 15*(b^3*cosh(x)^10
+ 10*b^3*cosh(x)*sinh(x)^9 + b^3*sinh(x)^10 - 5*b^3*cosh(x)^8 + 10*b^3*cosh(x)^6 + 5*(9*b^3*cosh(x)^2 - b^3)*
sinh(x)^8 + 40*(3*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^7 - 10*b^3*cosh(x)^4 + 10*(21*b^3*cosh(x)^4 - 14*b^3*co
sh(x)^2 + b^3)*sinh(x)^6 + 4*(63*b^3*cosh(x)^5 - 70*b^3*cosh(x)^3 + 15*b^3*cosh(x))*sinh(x)^5 + 5*b^3*cosh(x)^
2 + 10*(21*b^3*cosh(x)^6 - 35*b^3*cosh(x)^4 + 15*b^3*cosh(x)^2 - b^3)*sinh(x)^4 + 40*(3*b^3*cosh(x)^7 - 7*b^3*
cosh(x)^5 + 5*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^3 - b^3 + 5*(9*b^3*cosh(x)^8 - 28*b^3*cosh(x)^6 + 30*b^3*co
sh(x)^4 - 12*b^3*cosh(x)^2 + b^3)*sinh(x)^2 + 10*(b^3*cosh(x)^9 - 4*b^3*cosh(x)^7 + 6*b^3*cosh(x)^5 - 4*b^3*co
sh(x)^3 + b^3*cosh(x))*sinh(x))*sqrt(a^2 + a*b)*log((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 +
2*(2*a*b + b^2)*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + 2*a*b + b^2)*sinh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*cosh(x
)^3 + (2*a*b + b^2)*cosh(x))*sinh(x) + 4*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*a + b)*sqrt(a^2
+ a*b))/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a +
b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 80*(6*(a^2*b^2 + a*b^3)*cosh(x)^7 - 9*(a^3*
b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^5 + 2*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^3 - (4*a^4 + 17*a^3*
b + 28*a^2*b^2 + 15*a*b^3)*cosh(x))*sinh(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^10 + 10*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)*sinh(x)^9 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a
*b^4)*sinh(x)^10 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2
+ 4*a^2*b^3 + a*b^4 - 9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^5 + 4*
a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*si
nh(x)^7 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2
*b^3 + a*b^4 + 21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 14*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 70
*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4
)*cosh(x))*sinh(x)^5 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2*b^3 - a*b^4 - 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b
^3 + a*b^4)*cosh(x)^4 + 10*(21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 - a^5 - 4*a^4*b - 6*a
^3*b^2 - 4*a^2*b^3 - a*b^4 - 35*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 + 15*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*
cosh(x)^7 - 7*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2
*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^3 + 5*(a^5 + 4*a^4*
b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2 + 5*(9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8
- 28*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4
+ 30*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 12*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 +
a*b^4)*cosh(x)^2)*sinh(x)^2 + 10*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^9 - 4*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^7 + 6*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 4*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*co
sh(x))*sinh(x)), -1/15*(30*(a^2*b^2 + a*b^3)*cosh(x)^8 + 240*(a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + 30*(a^2*b^2
+ a*b^3)*sinh(x)^8 - 60*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^6 - 60*(a^3*b + 4*a^2*b^2 + 3*a*b^3 - 14*(a^2*b
^2 + a*b^3)*cosh(x)^2)*sinh(x)^6 + 120*(14*(a^2*b^2 + a*b^3)*cosh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(
x))*sinh(x)^5 + 20*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^4 + 20*(105*(a^2*b^2 + a*b^3)*cosh(x)^4
+ 8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^2)*sinh(x)^4 + 16*a^4 +
68*a^3*b + 118*a^2*b^2 + 66*a*b^3 + 80*(21*(a^2*b^2 + a*b^3)*cosh(x)^5 - 15*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh
(x)^3 + (8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x))*sinh(x)^3 - 20*(4*a^4 + 17*a^3*b + 28*a^2*b^2 + 15
*a*b^3)*cosh(x)^2 + 20*(42*(a^2*b^2 + a*b^3)*cosh(x)^6 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^4 - 4*a^4 -
17*a^3*b - 28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^2)*sinh(x)^2 + 15*(b^3
*cosh(x)^10 + 10*b^3*cosh(x)*sinh(x)^9 + b^3*sinh(x)^10 - 5*b^3*cosh(x)^8 + 10*b^3*cosh(x)^6 + 5*(9*b^3*cosh(x
)^2 - b^3)*sinh(x)^8 + 40*(3*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^7 - 10*b^3*cosh(x)^4 + 10*(21*b^3*cosh(x)^4
- 14*b^3*cosh(x)^2 + b^3)*sinh(x)^6 + 4*(63*b^3*cosh(x)^5 - 70*b^3*cosh(x)^3 + 15*b^3*cosh(x))*sinh(x)^5 + 5*b
^3*cosh(x)^2 + 10*(21*b^3*cosh(x)^6 - 35*b^3*cosh(x)^4 + 15*b^3*cosh(x)^2 - b^3)*sinh(x)^4 + 40*(3*b^3*cosh(x)
^7 - 7*b^3*cosh(x)^5 + 5*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^3 - b^3 + 5*(9*b^3*cosh(x)^8 - 28*b^3*cosh(x)^6
+ 30*b^3*cosh(x)^4 - 12*b^3*cosh(x)^2 + b^3)*sinh(x)^2 + 10*(b^3*cosh(x)^9 - 4*b^3*cosh(x)^7 + 6*b^3*cosh(x)^5
- 4*b^3*cosh(x)^3 + b^3*cosh(x))*sinh(x))*sqrt(-a^2 - a*b)*arctan(1/2*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*
sinh(x)^2 + 2*a + b)*sqrt(-a^2 - a*b)/(a^2 + a*b)) + 40*(6*(a^2*b^2 + a*b^3)*cosh(x)^7 - 9*(a^3*b + 4*a^2*b^2
+ 3*a*b^3)*cosh(x)^5 + 2*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^3 - (4*a^4 + 17*a^3*b + 28*a^2*b^2
+ 15*a*b^3)*cosh(x))*sinh(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^10 + 10*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)*sinh(x)^9 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*sinh(x)^
10 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 +
a*b^4 - 9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*
b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^7 + 10*(
a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 +
21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 14*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*
b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 70*(a^5 + 4*a^4*
b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sin
h(x)^5 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2*b^3 - a*b^4 - 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*co
sh(x)^4 + 10*(21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2
*b^3 - a*b^4 - 35*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^7 - 7*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*
cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^3 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2
+ 4*a^2*b^3 + a*b^4)*cosh(x)^2 + 5*(9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 28*(a^5 + 4*
a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + 30*(a^5 + 4
*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 12*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)
^2)*sinh(x)^2 + 10*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^9 - 4*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^7 + 6*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 4*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)
)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)**6/(a+b*cosh(x)**2),x)
[Out]
Timed out
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Giac [B] time = 1.41892, size = 255, normalized size = 2.87 \begin{align*} -\frac{b^{3} \arctan \left (\frac{b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt{-a^{2} - a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{-a^{2} - a b}} - \frac{2 \,{\left (15 \, b^{2} e^{\left (8 \, x\right )} - 30 \, a b e^{\left (6 \, x\right )} - 90 \, b^{2} e^{\left (6 \, x\right )} + 80 \, a^{2} e^{\left (4 \, x\right )} + 230 \, a b e^{\left (4 \, x\right )} + 240 \, b^{2} e^{\left (4 \, x\right )} - 40 \, a^{2} e^{\left (2 \, x\right )} - 130 \, a b e^{\left (2 \, x\right )} - 150 \, b^{2} e^{\left (2 \, x\right )} + 8 \, a^{2} + 26 \, a b + 33 \, b^{2}\right )}}{15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="giac")
[Out]
-b^3*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a^2 - a*b)) - 2
/15*(15*b^2*e^(8*x) - 30*a*b*e^(6*x) - 90*b^2*e^(6*x) + 80*a^2*e^(4*x) + 230*a*b*e^(4*x) + 240*b^2*e^(4*x) - 4
0*a^2*e^(2*x) - 130*a*b*e^(2*x) - 150*b^2*e^(2*x) + 8*a^2 + 26*a*b + 33*b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
(e^(2*x) - 1)^5)