3.1.41 \(\int \frac {\text {csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [41]

Optimal. Leaf size=110 \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a-b} d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3266, 472, 214} \begin {gather*} -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} d \sqrt {a-b}}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}-\frac {\coth ^5(c+d x)}{5 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

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 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 3266

Int[sin[(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^(m + 1)/f, Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p +
1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\text {csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^6}+\frac {-2 a-b}{a^2 x^4}+\frac {a^2+a b+b^2}{a^3 x^2}+\frac {b^3}{a^3 \left (-a+(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-a+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a^3 d}\\ &=-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a-b} d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 155, normalized size = 1.41 \begin {gather*} -\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \left (15 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \sqrt {a-b} \coth (c+d x) \left (8 a^2+10 a b+15 b^2-a (4 a+5 b) \text {csch}^2(c+d x)+3 a^2 \text {csch}^4(c+d x)\right )\right )}{30 a^{7/2} \sqrt {a-b} d \left (b+a \text {csch}^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

-1/30*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(15*b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] +
Sqrt[a]*Sqrt[a - b]*Coth[c + d*x]*(8*a^2 + 10*a*b + 15*b^2 - a*(4*a + 5*b)*Csch[c + d*x]^2 + 3*a^2*Csch[c + d*
x]^4)))/(a^(7/2)*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(98)=196\).
time = 1.58, size = 347, normalized size = 3.15

method	result	size
risch	\(-\frac {2 \left (15 b^{2} {\mathrm e}^{8 d x +8 c}-30 a b \,{\mathrm e}^{6 d x +6 c}-60 b^{2} {\mathrm e}^{6 d x +6 c}+80 a^{2} {\mathrm e}^{4 d x +4 c}+70 a b \,{\mathrm e}^{4 d x +4 c}+90 b^{2} {\mathrm e}^{4 d x +4 c}-40 a^{2} {\mathrm e}^{2 d x +2 c}-50 a b \,{\mathrm e}^{2 d x +2 c}-60 b^{2} {\mathrm e}^{2 d x +2 c}+8 a^{2}+10 a b +15 b^{2}\right )}{15 d \,a^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, d \,a^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, d \,a^{3}}\)	\(325\)
derivativedivides	\(\frac {-\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {4 a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{3}+10 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+12 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b +16 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{3}}+\frac {2 b^{3} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a^{2}}-\frac {1}{160 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-5 a -4 b}{96 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {10 a^{2}+12 a b +16 b^{2}}{32 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(347\)
default	\(\frac {-\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {4 a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{3}+10 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+12 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b +16 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{3}}+\frac {2 b^{3} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a^{2}}-\frac {1}{160 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-5 a -4 b}{96 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {10 a^{2}+12 a b +16 b^{2}}{32 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^6/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/32/a^3*(1/5*a^2*tanh(1/2*d*x+1/2*c)^5-5/3*a^2*tanh(1/2*d*x+1/2*c)^3-4/3*a*tanh(1/2*d*x+1/2*c)^3*b+10*a
^2*tanh(1/2*d*x+1/2*c)+12*a*tanh(1/2*d*x+1/2*c)*b+16*b^2*tanh(1/2*d*x+1/2*c))+2*b^3/a^2*(1/2*((-b*(a-b))^(1/2)
+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-
a+2*b)*a)^(1/2))-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*ta
nh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)))-1/160/a/tanh(1/2*d*x+1/2*c)^5-1/96*(-5*a-4*b)/a^2/tan
h(1/2*d*x+1/2*c)^3-1/32/a^3*(10*a^2+12*a*b+16*b^2)/tanh(1/2*d*x+1/2*c))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2142 vs. \(2 (98) = 196\).
time = 0.48, size = 4540, normalized size = 41.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/30*(60*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 480*(a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + 60*(a^2*b
^2 - a*b^3)*sinh(d*x + c)^8 - 120*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 120*(a^3*b + a^2*b^2 - 2*a*b^3
- 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 240*(14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^3 - 3*(a^3*
b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 40*(8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)
^4 + 40*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3 - 45*(a^3*b + a^2*b^2 - 2
*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*a^4 + 8*a^3*b + 20*a^2*b^2 - 60*a*b^3 + 160*(21*(a^2*b^2 - a*b^3
)*cosh(d*x + c)^5 - 15*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cos
h(d*x + c))*sinh(d*x + c)^3 - 40*(4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*cosh(d*x + c)^2 + 40*(42*(a^2*b^2 - a*b^3
)*cosh(d*x + c)^6 - 45*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 - 4*a^4 - a^3*b - a^2*b^2 + 6*a*b^3 + 6*(8*
a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 15*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d
*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 - 5*b^3*cosh(d*x + c)^8 + 10*b^3*cosh(d*x + c)^6 + 5*(9*b^3*cos
h(d*x + c)^2 - b^3)*sinh(d*x + c)^8 + 40*(3*b^3*cosh(d*x + c)^3 - b^3*cosh(d*x + c))*sinh(d*x + c)^7 - 10*b^3*
cosh(d*x + c)^4 + 10*(21*b^3*cosh(d*x + c)^4 - 14*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(
d*x + c)^5 - 70*b^3*cosh(d*x + c)^3 + 15*b^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^2 + 10*(21*b
^3*cosh(d*x + c)^6 - 35*b^3*cosh(d*x + c)^4 + 15*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^4 + 40*(3*b^3*cosh(d
*x + c)^7 - 7*b^3*cosh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^3 - b^3*cosh(d*x + c))*sinh(d*x + c)^3 - b^3 + 5*(9*b^
3*cosh(d*x + c)^8 - 28*b^3*cosh(d*x + c)^6 + 30*b^3*cosh(d*x + c)^4 - 12*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x +
c)^2 + 10*(b^3*cosh(d*x + c)^9 - 4*b^3*cosh(d*x + c)^7 + 6*b^3*cosh(d*x + c)^5 - 4*b^3*cosh(d*x + c)^3 + b^3*
cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 +
b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)
^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(b*cosh(d*x
+ c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 +
4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2
+ 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 80*(6*(a^2*
b^2 - a*b^3)*cosh(d*x + c)^7 - 9*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + 2*(8*a^4 - a^3*b + 2*a^2*b^2 -
9*a*b^3)*cosh(d*x + c)^3 - (4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 - a^4*b)*d*
cosh(d*x + c)^10 + 10*(a^5 - a^4*b)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^5 - a^4*b)*d*sinh(d*x + c)^10 - 5*(a^
5 - a^4*b)*d*cosh(d*x + c)^8 + 5*(9*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)^8 + 10*(a
^5 - a^4*b)*d*cosh(d*x + c)^6 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*
x + c)^7 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 14*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + (a^5 - a^4*b)*d)*sinh
(d*x + c)^6 - 10*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 4*(63*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 70*(a^5 - a^4*b)*d*
cosh(d*x + c)^3 + 15*(a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c)^6 -
35*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 15*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)^4 +
5*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^7 - 7*(a^5 - a^4*b)*d*cosh(d*x + c)^5
+ 5*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*(a^5 - a^4*b)*d*co
sh(d*x + c)^8 - 28*(a^5 - a^4*b)*d*cosh(d*x + c)^6 + 30*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 12*(a^5 - a^4*b)*d*c
osh(d*x + c)^2 + (a^5 - a^4*b)*d)*sinh(d*x + c)^2 - (a^5 - a^4*b)*d + 10*((a^5 - a^4*b)*d*cosh(d*x + c)^9 - 4*
(a^5 - a^4*b)*d*cosh(d*x + c)^7 + 6*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 4*(a^5 - a^4*b)*d*cosh(d*x + c)^3 + (a^5
- a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)), -1/15*(30*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 240*(a^2*b^2 - a*b^3)
*cosh(d*x + c)*sinh(d*x + c)^7 + 30*(a^2*b^2 - a*b^3)*sinh(d*x + c)^8 - 60*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*
x + c)^6 - 60*(a^3*b + a^2*b^2 - 2*a*b^3 - 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 120*(14*(a^
2*b^2 - a*b^3)*cosh(d*x + c)^3 - 3*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 20*(8*a^4 - a^
3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^4 + 20*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - a^3*b + 2*a^2
*b^2 - 9*a*b^3 - 45*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*a^4 + 4*a^3*b + 10*a^2*b
^2 - 30*a*b^3 + 80*(21*(a^2*b^2 - a*b^3)*cosh(d*x + c)^5 - 15*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (8
*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**6/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (98) = 196\).
time = 0.69, size = 213, normalized size = 1.94 \begin {gather*} -\frac {\frac {15 \, b^{3} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a^{3}} + \frac {2 \, {\left (15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 60 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 50 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 60 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )}}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/15*(15*b^3*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a^3) + 2*(15*b^2*e^
(8*d*x + 8*c) - 30*a*b*e^(6*d*x + 6*c) - 60*b^2*e^(6*d*x + 6*c) + 80*a^2*e^(4*d*x + 4*c) + 70*a*b*e^(4*d*x + 4
*c) + 90*b^2*e^(4*d*x + 4*c) - 40*a^2*e^(2*d*x + 2*c) - 50*a*b*e^(2*d*x + 2*c) - 60*b^2*e^(2*d*x + 2*c) + 8*a^
2 + 10*a*b + 15*b^2)/(a^3*(e^(2*d*x + 2*c) - 1)^5))/d

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Mupad [B]
time = 1.21, size = 479, normalized size = 4.35 \begin {gather*} \frac {4\,b}{a^2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {32}{5\,a\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {2\,b^2}{a^3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (4\,a-b\right )}{3\,a^2\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {16}{a\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^7\,\left (a-b\right )}-\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{13/2}\,\sqrt {a-b}}\right )}{2\,a^{7/2}\,d\,\sqrt {a-b}}-\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^7\,\left (a-b\right )}+\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{13/2}\,\sqrt {a-b}}\right )}{2\,a^{7/2}\,d\,\sqrt {a-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^6*(a + b*sinh(c + d*x)^2)),x)

[Out]

(4*b)/(a^2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - 32/(5*a*d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x
) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1)) - (2*b^2)/(a^3*d*(exp(2*c + 2*d*x) - 1
)) - (8*(4*a - b))/(3*a^2*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - 16/(a*d*(6*exp
(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) + (b^3*log((4*b^4*(2*a*b - b^
2 + 8*a^2*exp(2*c + 2*d*x) + b^2*exp(2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a^7*(a - b)) - (8*b^4*(b + 4*a*e
xp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(13/2)*(a - b)^(1/2))))/(2*a^(7/2)*d*(a - b)^(1/2)) - (b^3*log((4*
b^4*(2*a*b - b^2 + 8*a^2*exp(2*c + 2*d*x) + b^2*exp(2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a^7*(a - b)) + (8
*b^4*(b + 4*a*exp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(13/2)*(a - b)^(1/2))))/(2*a^(7/2)*d*(a - b)^(1/2))

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