3.1.50 \(\int \frac {\text {csch}^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [50]

Optimal. Leaf size=174 \[ \frac {(6 a-5 b) b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} (a-b)^{3/2} d}+\frac {\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a-b) d}-\frac {(2 a-5 b) \coth ^3(c+d x)}{6 a^2 (a-b) d}-\frac {b \text {csch}^3(c+d x) \text {sech}(c+d x)}{2 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

1/2*(6*a-5*b)*b^2*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(7/2)/(a-b)^(3/2)/d+1/2*(2*a^2+a*b-5*b^2)*coth(d*
x+c)/a^3/(a-b)/d-1/6*(2*a-5*b)*coth(d*x+c)^3/a^2/(a-b)/d-1/2*b*csch(d*x+c)^3*sech(d*x+c)/a/(a-b)/d/(a-(a-b)*ta
nh(d*x+c)^2)

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Rubi [A]
time = 0.17, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3266, 479, 584, 214} \begin {gather*} \frac {b^2 (6 a-5 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} d (a-b)^{3/2}}-\frac {(2 a-5 b) \coth ^3(c+d x)}{6 a^2 d (a-b)}+\frac {\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 d (a-b)}-\frac {b \text {csch}^3(c+d x) \text {sech}(c+d x)}{2 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^(3/2)*d) + ((2*a^2 + a*b - 5
*b^2)*Coth[c + d*x])/(2*a^3*(a - b)*d) - ((2*a - 5*b)*Coth[c + d*x]^3)/(6*a^2*(a - b)*d) - (b*Csch[c + d*x]^3*
Sech[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))

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 479

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

Rule 584

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

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

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Mathematica [A]
time = 0.83, size = 210, normalized size = 1.21 \begin {gather*} \frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^4(c+d x) \left (\frac {3 (6 a-5 b) b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right ) (2 a-b+b \cosh (2 (c+d x)))}{(a-b)^{3/2}}+4 \sqrt {a} (a+3 b) (2 a-b+b \cosh (2 (c+d x))) \coth (c+d x)-2 a^{3/2} (2 a-b+b \cosh (2 (c+d x))) \coth (c+d x) \text {csch}^2(c+d x)-\frac {3 \sqrt {a} b^3 \sinh (2 (c+d x))}{a-b}\right )}{24 a^{7/2} d \left (b+a \text {csch}^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*((3*(6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[
a]]*(2*a - b + b*Cosh[2*(c + d*x)]))/(a - b)^(3/2) + 4*Sqrt[a]*(a + 3*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[
c + d*x] - 2*a^(3/2)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x]^2 - (3*Sqrt[a]*b^3*Sinh[2*(c
+ d*x)])/(a - b)))/(24*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(158)=316\).
time = 1.61, size = 371, normalized size = 2.13

method	result	size
derivativedivides	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}}-\frac {1}{24 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-8 b -3 a}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b^{2} \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a -2 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (6 a -5 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{a^{3}}}{d}\)	\(371\)
default	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}}-\frac {1}{24 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-8 b -3 a}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b^{2} \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a -2 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (6 a -5 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{a^{3}}}{d}\)	\(371\)
risch	\(-\frac {-18 a \,b^{2} {\mathrm e}^{8 d x +8 c}+15 b^{3} {\mathrm e}^{8 d x +8 c}-36 a^{2} b \,{\mathrm e}^{6 d x +6 c}+102 a \,b^{2} {\mathrm e}^{6 d x +6 c}-60 b^{3} {\mathrm e}^{6 d x +6 c}+48 a^{3} {\mathrm e}^{4 d x +4 c}+20 a^{2} b \,{\mathrm e}^{4 d x +4 c}-158 a \,b^{2} {\mathrm e}^{4 d x +4 c}+90 b^{3} {\mathrm e}^{4 d x +4 c}-16 a^{3} {\mathrm e}^{2 d x +2 c}-12 a^{2} b \,{\mathrm e}^{2 d x +2 c}+82 a \,b^{2} {\mathrm e}^{2 d x +2 c}-60 b^{3} {\mathrm e}^{2 d x +2 c}-4 a^{2} b -8 a \,b^{2}+15 b^{3}}{3 a^{3} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left (a -b \right ) \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}-\frac {5 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{3}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}+\frac {5 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{3}}\)	\(632\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8/a^3*(1/3*a*tanh(1/2*d*x+1/2*c)^3-3*a*tanh(1/2*d*x+1/2*c)-8*b*tanh(1/2*d*x+1/2*c))-1/24/a^2/tanh(1/2*
d*x+1/2*c)^3-1/8/a^3*(-8*b-3*a)/tanh(1/2*d*x+1/2*c)-2*b^2/a^3*((1/2*b/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/2*b/(a-b)*
tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/2*(6*a-
5*b)/(a-b)*a*(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*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)))))

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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)^4/(a+b*sinh(d*x+c)^2)^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. 3427 vs. \(2 (159) = 318\).
time = 0.47, size = 7110, normalized size = 40.86 \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)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/12*(12*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^8 + 96*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x
+ c)*sinh(d*x + c)^7 + 12*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*sinh(d*x + c)^8 + 24*(6*a^4*b - 23*a^3*b^2 + 27*a
^2*b^3 - 10*a*b^4)*cosh(d*x + c)^6 + 24*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4 + 14*(6*a^3*b^2 - 11*a^2
*b^3 + 5*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 48*(14*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^3 +
3*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 16*a^4*b + 16*a^3*b^2 - 92*
a^2*b^3 + 60*a*b^4 - 8*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^4 - 8*(24*a^5 -
14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4 - 105*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^4 - 45*
(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(6*a^3*b^2 - 11*a^2*b
^3 + 5*a*b^4)*cosh(d*x + c)^5 + 15*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^3 - (24*a^5 -
14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(8*a^5 - 2*a^4*b - 47*a^3*b
^2 + 71*a^2*b^3 - 30*a*b^4)*cosh(d*x + c)^2 + 8*(42*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^6 + 8*a^5
- 2*a^4*b - 47*a^3*b^2 + 71*a^2*b^3 - 30*a*b^4 + 45*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x +
c)^4 - 6*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((6*a
*b^3 - 5*b^4)*cosh(d*x + c)^10 + 10*(6*a*b^3 - 5*b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (6*a*b^3 - 5*b^4)*sinh(d
*x + c)^10 + (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^8 + (24*a^2*b^2 - 50*a*b^3 + 25*b^4 + 45*(6*a*b^3
- 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^3 + (24*a^2*b^2 - 50*a*b^3 +
25*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^6 + 2*(105*(6*a*b^3
- 5*b^4)*cosh(d*x + c)^4 - 36*a^2*b^2 + 60*a*b^3 - 25*b^4 + 14*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 4*(63*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 14*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x +
c)^3 - 3*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)
*cosh(d*x + c)^4 + 2*(105*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 35*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c
)^4 + 36*a^2*b^2 - 60*a*b^3 + 25*b^4 - 15*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 -
6*a*b^3 + 5*b^4 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 7*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^5
- 5*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 + (36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d
*x + c)^3 - (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^2 + (45*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^8 + 28*(24*
a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^6 - 30*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^4 - 24*a^2*b^
2 + 50*a*b^3 - 25*b^4 + 12*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(6*a*b^3 -
5*b^4)*cosh(d*x + c)^9 + 4*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^7 - 6*(36*a^2*b^2 - 60*a*b^3 + 25*b
^4)*cosh(d*x + c)^5 + 4*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 - (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*co
sh(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)) + 16*(6*(6*a^3*
b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^7 + 9*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^5
- 2*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^3 + (8*a^5 - 2*a^4*b - 47*a^3*b^2
+ 71*a^2*b^3 - 30*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^10 + 10
*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6*b - 2*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)
^10 + (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^8 + (45*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh
(d*x + c)^2 + (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^8 - 2*(6*a^7 - 17*a^6*b + 16*a^5*b^
2 - 5*a^4*b^3)*d*cosh(d*x + c)^6 + 8*(15*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + (4*a^7 - 13*a^6*b +
14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x +
c)^4 + 14*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 - (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a
^4*b^3)*d)*sinh(d*x + c)^6 + 2*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 + 4*(63*(a^6*b -
2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 + 14*(4*...

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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)**4/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.72, size = 220, normalized size = 1.26 \begin {gather*} \frac {\frac {3 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{4} - a^{3} b\right )} \sqrt {-a^{2} + a b}} + \frac {6 \, {\left (2 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )}}{{\left (a^{4} - a^{3} b\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}} + \frac {8 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + 3 \, b\right )}}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(6*a*b^2 - 5*b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^4 - a^3*b)*sqrt(-a^2 +
a*b)) + 6*(2*a*b^2*e^(2*d*x + 2*c) - b^3*e^(2*d*x + 2*c) + b^3)/((a^4 - a^3*b)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*
d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)) + 8*(3*b*e^(4*d*x + 4*c) - 3*a*e^(2*d*x + 2*c) - 6*b*e^(2*d*x + 2*c) +
a + 3*b)/(a^3*(e^(2*d*x + 2*c) - 1)^3))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^4*(a + b*sinh(c + d*x)^2)^2),x)

[Out]

int(1/(sinh(c + d*x)^4*(a + b*sinh(c + d*x)^2)^2), x)

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