3.3.64 \(\int \frac {\text {csch}^2(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [264]

Optimal. Leaf size=359 \[ -\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}+\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]

[Out]

-coth(d*x+c)/a^3/d-3/64*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*b^(1/2)*(20*a+15*b-34*a^(1/2)*b^(
1/2))/a^(13/4)/d/(a^(1/2)-b^(1/2))^(5/2)+3/64*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*b^(1/2)*(20
*a+15*b+34*a^(1/2)*b^(1/2))/a^(13/4)/d/(a^(1/2)+b^(1/2))^(5/2)+1/8*b^2*tanh(d*x+c)*(a*(a+3*b)-(a^2+6*a*b+b^2)*
tanh(d*x+c)^2)/a^2/(a-b)^3/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)^2+1/32*b*tanh(d*x+c)*(2*a^2*(9*a-17*b)/
(a-b)^3-(18*a^2+15*a*b-13*b^2)*tanh(d*x+c)^2/(a-b)^2)/a^3/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)

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Rubi [A]
time = 0.84, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 1348, 1683, 1678, 1180, 214} \begin {gather*} -\frac {3 \sqrt {b} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {3 \sqrt {b} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 d (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

(-3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a
^(13/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt[a] + S
qrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - Coth[c + d*x]/(a^3*d) + (b^2*Tanh
[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (
a - b)*Tanh[c + d*x]^4)^2) + (b*Tanh[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 - ((18*a^2 + 15*a*b - 13*b^2)*Ta
nh[c + d*x]^2)/(a - b)^2))/(32*a^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))

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 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1348

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d +
e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*
f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*
x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x
^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x],
x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]

Rule 1678

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1683

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rule 3296

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 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}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^6}{x^2 \left (a-2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-16 a b+\frac {2 a b \left (32 a^3-96 a^2 b+97 a b^2-29 b^3\right ) x^2}{(a-b)^3}-\frac {2 b \left (48 a^4-136 a^3 b+115 a^2 b^2-30 a b^3-5 b^4\right ) x^4}{(a-b)^3}+\frac {32 a^2 (2 a-3 b) b x^6}{(a-b)^2}-\frac {16 a^2 b x^8}{a-b}}{x^2 \left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{16 a^2 b d}\\ &=\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {128 a^2 b^2-\frac {8 a^2 b^2 \left (32 a^2-55 a b+26 b^2\right ) x^2}{(a-b)^2}+\frac {4 a b^2 \left (32 a^3-18 a^2 b-15 a b^2+13 b^3\right ) x^4}{(a-b)^2}}{x^2 \left (a-2 a x^2+(a-b) x^4\right )} \, dx,x,\tanh (c+d x)\right )}{128 a^4 b^2 d}\\ &=\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \left (\frac {128 a b^2}{x^2}+\frac {12 a b^3 \left (-2 a (3 a-2 b)+\left (26 a^2-37 a b+15 b^2\right ) x^2\right )}{(a-b)^2 \left (a-2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{128 a^4 b^2 d}\\ &=-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {(3 b) \text {Subst}\left (\int \frac {-2 a (3 a-2 b)+\left (26 a^2-37 a b+15 b^2\right ) x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{32 a^3 (a-b)^2 d}\\ &=-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {\left (3 \left (\sqrt {a}+\sqrt {b}\right )^3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right )\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{64 a^3 (a-b)^2 d}-\frac {\left (3 \left (\sqrt {a}-\sqrt {b}\right )^3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right )\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{64 a^3 (a-b)^2 d}\\ &=-\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}+\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 2.41, size = 357, normalized size = 0.99 \begin {gather*} \frac {\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {a+\sqrt {a} \sqrt {b}}}-64 \coth (c+d x)+\frac {4 b \left (28 a^2+3 a b-13 b^2+b (-19 a+13 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b)^2 (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {128 a b (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b) (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{64 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sq
rt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*Ar
cTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*
Sqrt[b]]) - 64*Coth[c + d*x] + (4*b*(28*a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cosh[2*(c + d*x)])*Sinh[2*(c +
d*x)])/((a - b)^2*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cosh[2*(
c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/(64*a^3*
d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.18, size = 608, normalized size = 1.69

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {8 b \left (\frac {-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{3}-50 a^{2} b -612 a \,b^{2}+416 b^{3}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {\left (45 a^{3}-50 a^{2} b -612 a \,b^{2}+416 b^{3}\right ) \left (\tanh ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \left (\tanh ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \left (\tanh ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right ) \left (\tanh ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (-3 a +2 b \right ) \textit {\_R}^{6}+\left (49 a^{2}-72 a b +30 b^{2}\right ) \textit {\_R}^{4}+\left (-49 a^{2}+72 a b -30 b^{2}\right ) \textit {\_R}^{2}+3 a^{2}-2 a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{512 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(608\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {8 b \left (\frac {-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{3}-50 a^{2} b -612 a \,b^{2}+416 b^{3}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {\left (45 a^{3}-50 a^{2} b -612 a \,b^{2}+416 b^{3}\right ) \left (\tanh ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \left (\tanh ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \left (\tanh ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right ) \left (\tanh ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (-3 a +2 b \right ) \textit {\_R}^{6}+\left (49 a^{2}-72 a b +30 b^{2}\right ) \textit {\_R}^{4}+\left (-49 a^{2}+72 a b -30 b^{2}\right ) \textit {\_R}^{2}+3 a^{2}-2 a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{512 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(608\)
risch	\(\text {Expression too large to display}\)	\(2723\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a^3*tanh(1/2*d*x+1/2*c)-8/a^3*b*((-3/64*a^2*(3*a-2*b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+1/64*(45*a
^2+16*a*b-34*b^2)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/64*a*(81*a^2-28*a*b-38*b^2)/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^5+1/64*(45*a^3-50*a^2*b-612*a*b^2+416*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+1/64*(45*a^3-50
*a^2*b-612*a*b^2+416*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-1/64*a*(81*a^2-28*a*b-38*b^2)/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)^11+1/64*(45*a^2+16*a*b-34*b^2)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^13-3/64*a^2*(3*a-2*b)
/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^15)/(a*tanh(1/2*d*x+1/2*c)^8-4*a*tanh(1/2*d*x+1/2*c)^6+6*a*tanh(1/2*d*x+1
/2*c)^4-16*b*tanh(1/2*d*x+1/2*c)^4-4*a*tanh(1/2*d*x+1/2*c)^2+a)^2+3/512/(a^2-2*a*b+b^2)*sum((a*(-3*a+2*b)*_R^6
+(49*a^2-72*a*b+30*b^2)*_R^4+(-49*a^2+72*a*b-30*b^2)*_R^2+3*a^2-2*a*b)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a
)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a)))-1/2/a^3/tanh(1/2*d*x+1/2*
c))

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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(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/16*(32*a^2*b^2 - 83*a*b^3 + 45*b^4 + 3*(20*a^2*b^2*e^(16*c) - 33*a*b^3*e^(16*c) + 15*b^4*e^(16*c))*e^(16*d*x
) - 12*(43*a^2*b^2*e^(14*c) - 68*a*b^3*e^(14*c) + 30*b^4*e^(14*c))*e^(14*d*x) - 4*(400*a^3*b*e^(12*c) - 1137*a
^2*b^2*e^(12*c) + 1031*a*b^3*e^(12*c) - 315*b^4*e^(12*c))*e^(12*d*x) + 12*(592*a^3*b*e^(10*c) - 1237*a^2*b^2*e
^(10*c) + 886*a*b^3*e^(10*c) - 210*b^4*e^(10*c))*e^(10*d*x) + 2*(4096*a^4*e^(8*c) - 12192*a^3*b*e^(8*c) + 1363
4*a^2*b^2*e^(8*c) - 7113*a*b^3*e^(8*c) + 1575*b^4*e^(8*c))*e^(8*d*x) + 4*(880*a^3*b*e^(6*c) - 2855*a^2*b^2*e^(
6*c) + 2512*a*b^3*e^(6*c) - 630*b^4*e^(6*c))*e^(6*d*x) - 4*(256*a^3*b*e^(4*c) - 823*a^2*b^2*e^(4*c) + 903*a*b^
3*e^(4*c) - 315*b^4*e^(4*c))*e^(4*d*x) - 12*(19*a^2*b^2*e^(2*c) - 54*a*b^3*e^(2*c) + 30*b^4*e^(2*c))*e^(2*d*x)
)/(a^5*b^2*d - 2*a^4*b^3*d + a^3*b^4*d - (a^5*b^2*d*e^(18*c) - 2*a^4*b^3*d*e^(18*c) + a^3*b^4*d*e^(18*c))*e^(1
8*d*x) + 9*(a^5*b^2*d*e^(16*c) - 2*a^4*b^3*d*e^(16*c) + a^3*b^4*d*e^(16*c))*e^(16*d*x) + 4*(8*a^6*b*d*e^(14*c)
- 25*a^5*b^2*d*e^(14*c) + 26*a^4*b^3*d*e^(14*c) - 9*a^3*b^4*d*e^(14*c))*e^(14*d*x) - 4*(40*a^6*b*d*e^(12*c) -
101*a^5*b^2*d*e^(12*c) + 82*a^4*b^3*d*e^(12*c) - 21*a^3*b^4*d*e^(12*c))*e^(12*d*x) - 2*(128*a^7*d*e^(10*c) -
416*a^6*b*d*e^(10*c) + 511*a^5*b^2*d*e^(10*c) - 286*a^4*b^3*d*e^(10*c) + 63*a^3*b^4*d*e^(10*c))*e^(10*d*x) + 2
*(128*a^7*d*e^(8*c) - 416*a^6*b*d*e^(8*c) + 511*a^5*b^2*d*e^(8*c) - 286*a^4*b^3*d*e^(8*c) + 63*a^3*b^4*d*e^(8*
c))*e^(8*d*x) + 4*(40*a^6*b*d*e^(6*c) - 101*a^5*b^2*d*e^(6*c) + 82*a^4*b^3*d*e^(6*c) - 21*a^3*b^4*d*e^(6*c))*e
^(6*d*x) - 4*(8*a^6*b*d*e^(4*c) - 25*a^5*b^2*d*e^(4*c) + 26*a^4*b^3*d*e^(4*c) - 9*a^3*b^4*d*e^(4*c))*e^(4*d*x)
- 9*(a^5*b^2*d*e^(2*c) - 2*a^4*b^3*d*e^(2*c) + a^3*b^4*d*e^(2*c))*e^(2*d*x)) - 4*integrate(3/32*((20*a^2*b*e^
(6*c) - 33*a*b^2*e^(6*c) + 15*b^3*e^(6*c))*e^(6*d*x) - 2*(32*a^2*b*e^(4*c) - 41*a*b^2*e^(4*c) + 15*b^3*e^(4*c)
)*e^(4*d*x) + (20*a^2*b*e^(2*c) - 33*a*b^2*e^(2*c) + 15*b^3*e^(2*c))*e^(2*d*x))/(a^5*b - 2*a^4*b^2 + a^3*b^3 +
(a^5*b*e^(8*c) - 2*a^4*b^2*e^(8*c) + a^3*b^3*e^(8*c))*e^(8*d*x) - 4*(a^5*b*e^(6*c) - 2*a^4*b^2*e^(6*c) + a^3*
b^3*e^(6*c))*e^(6*d*x) - 2*(8*a^6*e^(4*c) - 19*a^5*b*e^(4*c) + 14*a^4*b^2*e^(4*c) - 3*a^3*b^3*e^(4*c))*e^(4*d*
x) - 4*(a^5*b*e^(2*c) - 2*a^4*b^2*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28429 vs. \(2 (306) = 612\).
time = 1.48, size = 28429, normalized size = 79.19 \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)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/128*(24*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*cosh(d*x + c)^16 + 384*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*cosh(d*x +
c)*sinh(d*x + c)^15 + 24*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*sinh(d*x + c)^16 - 96*(43*a^2*b^2 - 68*a*b^3 + 30*b
^4)*cosh(d*x + c)^14 - 96*(43*a^2*b^2 - 68*a*b^3 + 30*b^4 - 30*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*cosh(d*x + c)^
2)*sinh(d*x + c)^14 + 1344*(10*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*cosh(d*x + c)^3 - (43*a^2*b^2 - 68*a*b^3 + 30*
b^4)*cosh(d*x + c))*sinh(d*x + c)^13 - 32*(400*a^3*b - 1137*a^2*b^2 + 1031*a*b^3 - 315*b^4)*cosh(d*x + c)^12 +
32*(1365*(20*a^2*b^2 - 33*a*b^3 + 15*b^4)*cosh(d*x + c)^4 - 400*a^3*b + 1137*a^2*b^2 - 1031*a*b^3 + 315*b^4 -
273*(43*a^2*b^2 - 68*a*b^3 + 30*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 384*(273*(20*a^2*b^2 - 33*a*b^3 + 15
*b^4)*cosh(d*x + c)^5 - 91*(43*a^2*b^2 - 68*a*b^3 + 30*b^4)*cosh(d*x + c)^3 - (400*a^3*b - 1137*a^2*b^2 + 1031
*a*b^3 - 315*b^4)*cosh(d*x + c))*sinh(d*x + c)^11 + 96*(592*a^3*b - 1237*a^2*b^2 + 886*a*b^3 - 210*b^4)*cosh(d
*x + c)^10 ...

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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)**2/(a-b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.59, size = 486, normalized size = 1.35 \begin {gather*} -\frac {\frac {28 \, a^{2} b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 35 \, a b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 13 \, b^{4} e^{\left (14 \, d x + 14 \, c\right )} - 232 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 269 \, a b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 91 \, b^{4} e^{\left (12 \, d x + 12 \, c\right )} - 576 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 1372 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 1039 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 273 \, b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 2432 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} - 3488 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1913 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 455 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 576 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 1060 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1689 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 455 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 376 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 679 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 273 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 117 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 91 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 19 \, a b^{3} - 13 \, b^{4}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} + \frac {32}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{16 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/16*((28*a^2*b^2*e^(14*d*x + 14*c) - 35*a*b^3*e^(14*d*x + 14*c) + 13*b^4*e^(14*d*x + 14*c) - 232*a^2*b^2*e^(
12*d*x + 12*c) + 269*a*b^3*e^(12*d*x + 12*c) - 91*b^4*e^(12*d*x + 12*c) - 576*a^3*b*e^(10*d*x + 10*c) + 1372*a
^2*b^2*e^(10*d*x + 10*c) - 1039*a*b^3*e^(10*d*x + 10*c) + 273*b^4*e^(10*d*x + 10*c) + 2432*a^3*b*e^(8*d*x + 8*
c) - 3488*a^2*b^2*e^(8*d*x + 8*c) + 1913*a*b^3*e^(8*d*x + 8*c) - 455*b^4*e^(8*d*x + 8*c) + 576*a^3*b*e^(6*d*x
+ 6*c) + 1060*a^2*b^2*e^(6*d*x + 6*c) - 1689*a*b^3*e^(6*d*x + 6*c) + 455*b^4*e^(6*d*x + 6*c) - 376*a^2*b^2*e^(
4*d*x + 4*c) + 679*a*b^3*e^(4*d*x + 4*c) - 273*b^4*e^(4*d*x + 4*c) - 28*a^2*b^2*e^(2*d*x + 2*c) - 117*a*b^3*e^
(2*d*x + 2*c) + 91*b^4*e^(2*d*x + 2*c) + 19*a*b^3 - 13*b^4)/((a^5 - 2*a^4*b + a^3*b^2)*(b*e^(8*d*x + 8*c) - 4*
b*e^(6*d*x + 6*c) - 16*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 4*c) - 4*b*e^(2*d*x + 2*c) + b)^2) + 32/(a^3*(e^(2*d
*x + 2*c) - 1)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^3),x)

[Out]

int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^3), x)

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