3.3.52 \(\int \frac {\text {csch}^2(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [252]

Optimal. Leaf size=237 \[ -\frac {\left (6 \sqrt {a}-5 \sqrt {b}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (6 \sqrt {a}+5 \sqrt {b}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {\coth (c+d x)}{a^2 d}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]

[Out]

-coth(d*x+c)/a^2/d-1/8*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(6*a^(1/2)-5*b^(1/2))*b^(1/2)/a^(9
/4)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*b^(1/2)*(6*a^(1/2)+5*b^
(1/2))/a^(9/4)/d/(a^(1/2)+b^(1/2))^(3/2)+1/4*b*tanh(d*x+c)*(a-(a+b)*tanh(d*x+c)^2)/a^2/(a-b)/d/(a-2*a*tanh(d*x
+c)^2+(a-b)*tanh(d*x+c)^4)

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1348, 1678, 1180, 214} \begin {gather*} -\frac {\sqrt {b} \left (6 \sqrt {a}-5 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt {b} \left (6 \sqrt {a}+5 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\coth (c+d x)}{a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(a^(9/4)*(Sqrt
[a] - Sqrt[b])^(3/2)*d) + ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(
1/4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Coth[c + d*x]/(a^2*d) + (b*Tanh[c + d*x]*(a - (a + b)*Tanh[c
+ d*x]^2))/(4*a^2*(a - b)*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 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 )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^2 \left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-8 a b+\frac {2 a (8 a-7 b) b x^2}{a-b}-\frac {2 b \left (4 a^2-a b-b^2\right ) x^4}{a-b}}{x^2 \left (a-2 a x^2+(a-b) x^4\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) 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 {8 b}{x^2}+\frac {2 b^2 \left (a-(7 a-5 b) x^2\right )}{(a-b) \left (a-2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=-\frac {\coth (c+d x)}{a^2 d}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {b \text {Subst}\left (\int \frac {a+(-7 a+5 b) x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{4 a^2 (a-b) d}\\ &=-\frac {\coth (c+d x)}{a^2 d}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {\left (\left (7 a+\frac {2 \sqrt {a} (3 a-2 b)}{\sqrt {b}}-5 b\right ) b\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b) d}-\frac {\left (b \left (-7 a+\frac {2 \sqrt {a} (3 a-2 b)}{\sqrt {b}}+5 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 )}{8 a^2 (a-b) d}\\ &=-\frac {\left (6 \sqrt {a}-5 \sqrt {b}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (6 \sqrt {a}+5 \sqrt {b}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {\coth (c+d x)}{a^2 d}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.40, size = 272, normalized size = 1.15 \begin {gather*} \frac {\frac {\left (6 a \sqrt {b}-5 \sqrt {a} 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 ) \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\left (6 a \sqrt {b}+5 \sqrt {a} 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 ) \sqrt {a+\sqrt {a} \sqrt {b}}}-8 \sqrt {a} \coth (c+d x)+\frac {4 \sqrt {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)))}}{8 a^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((6*a*Sqrt[b] - 5*Sqrt[a]*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a
] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((6*a*Sqrt[b] + 5*Sqrt[a]*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c +
d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) - 8*Sqrt[a]*Coth[c + d*x] +
(4*Sqrt[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)])))/(8*a^(5/2)*d)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.41, size = 326, normalized size = 1.38 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/2/a^2*tanh(1/2*d*x+1/2*c)-16*b/a^2*((-1/32*a/(a-b)*tanh(1/2*d*x+1/2*c)^7+1/32*(a+4*b)/(a-b)*tanh(1/2*d
*x+1/2*c)^5+1/32*(a+4*b)/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/32*a/(a-b)*tanh(1/2*d*x+1/2*c))/(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)+
1/256/(a-b)*sum((-a*_R^6+(27*a-20*b)*_R^4+(-27*a+20*b)*_R^2+a)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tan
h(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^2/tanh(1/2*d*x+1/2*c))

________________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

1/2*(4*a*b - 5*b^2 + (6*a*b*e^(8*c) - 5*b^2*e^(8*c))*e^(8*d*x) - 2*(13*a*b*e^(6*c) - 10*b^2*e^(6*c))*e^(6*d*x)
- 2*(32*a^2*e^(4*c) - 47*a*b*e^(4*c) + 15*b^2*e^(4*c))*e^(4*d*x) - 2*(7*a*b*e^(2*c) - 10*b^2*e^(2*c))*e^(2*d*
x))/(a^3*b*d - a^2*b^2*d - (a^3*b*d*e^(10*c) - a^2*b^2*d*e^(10*c))*e^(10*d*x) + 5*(a^3*b*d*e^(8*c) - a^2*b^2*d
*e^(8*c))*e^(8*d*x) + 2*(8*a^4*d*e^(6*c) - 13*a^3*b*d*e^(6*c) + 5*a^2*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^4*d*e^
(4*c) - 13*a^3*b*d*e^(4*c) + 5*a^2*b^2*d*e^(4*c))*e^(4*d*x) - 5*(a^3*b*d*e^(2*c) - a^2*b^2*d*e^(2*c))*e^(2*d*x
)) - 4*integrate(1/4*((6*a*b*e^(6*c) - 5*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a*b*e^(4*c) - 5*b^2*e^(4*c))*e^(4*d*x)
+ (6*a*b*e^(2*c) - 5*b^2*e^(2*c))*e^(2*d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(8*c) - a^2*b^2*e^(8*c))*e^(8*d*x) -
4*(a^3*b*e^(6*c) - a^2*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^4*e^(4*c) - 11*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c))*e^(4*
d*x) - 4*(a^3*b*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8824 vs. \(2 (188) = 376\).
time = 0.74, size = 8824, normalized size = 37.23 \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)^2,x, algorithm="fricas")

[Out]

-1/16*(8*(6*a*b - 5*b^2)*cosh(d*x + c)^8 + 64*(6*a*b - 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(6*a*b - 5*b^2
)*sinh(d*x + c)^8 - 16*(13*a*b - 10*b^2)*cosh(d*x + c)^6 + 16*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^2 - 13*a*b + 1
0*b^2)*sinh(d*x + c)^6 + 32*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^3 - 3*(13*a*b - 10*b^2)*cosh(d*x + c))*sinh(d*x
+ c)^5 - 16*(32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c)^4 + 16*(35*(6*a*b - 5*b^2)*cosh(d*x + c)^4 - 15*(13*a*b -
10*b^2)*cosh(d*x + c)^2 - 32*a^2 + 47*a*b - 15*b^2)*sinh(d*x + c)^4 + 64*(7*(6*a*b - 5*b^2)*cosh(d*x + c)^5 -
5*(13*a*b - 10*b^2)*cosh(d*x + c)^3 - (32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*(7*a*b -
10*b^2)*cosh(d*x + c)^2 + 16*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^6 - 15*(13*a*b - 10*b^2)*cosh(d*x + c)^4 - 6*(
32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c)^2 - 7*a*b + 10*b^2)*sinh(d*x + c)^2 + ((a^3*b - a^2*b^2)*d*cosh(d*x +
c)^10 + 10*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b
- a^2*b^2)*d*cosh(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^
8 - 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b -
a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)^2 - (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cos
h(d*x + c)^4 + 4*(63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 -
13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^
3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5
*a^2*b^2)*d)*sinh(d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)
^7 - 35*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13
*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b -
a^2*b^2)*d*cosh(d*x + c)^6 - 30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^
2*b^2)*d*cosh(d*x + c)^2 + 5*(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^
2)*d*cosh(d*x + c)^9 - 20*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x +
c)^5 + 4*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)
)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b
^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a
^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*
a*b^4 - 625*b^5 + 2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*sqrt((2304*a^4*b^3 - 662
4*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 -
6*a^10*b^5 + a^9*b^6)*d^4)) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)^2 - 2*(1728*
a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)*sinh(d*x + c) - (1728*a^3*b^2 - 3684*a^2*b^3 + 26
25*a*b^4 - 625*b^5)*sinh(d*x + c)^2 + 2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((
2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b
^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d)*sqr
t(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 +
625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b
+ 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cosh(d*x + c)^10 + 1
0*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b - a^2*b^
2)*d*cosh(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^8 - 2*(8*
a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*
d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*d*cosh(d*
x + c)^2 - (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c
)^4 + 4*(63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 - 13*a^3*b
+ 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^3*b - a^2
*b^2)*d*cosh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)
*d)*sinh(d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 35*(
a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 -...

________________________________________________________________________________________

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4373 deep

________________________________________________________________________________________

Giac [A]
time = 0.49, size = 238, normalized size = 1.00 \begin {gather*} -\frac {6 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 5 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 26 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 20 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 64 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 94 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 30 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 14 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 20 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a b - 5 \, b^{2}}{2 \, {\left (a^{3} - a^{2} b\right )} {\left (b e^{\left (10 \, d x + 10 \, c\right )} - 5 \, b e^{\left (8 \, d x + 8 \, c\right )} - 16 \, a e^{\left (6 \, d x + 6 \, c\right )} + 10 \, b e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a e^{\left (4 \, d x + 4 \, c\right )} - 10 \, b e^{\left (4 \, d x + 4 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} 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)^2,x, algorithm="giac")

[Out]

-1/2*(6*a*b*e^(8*d*x + 8*c) - 5*b^2*e^(8*d*x + 8*c) - 26*a*b*e^(6*d*x + 6*c) + 20*b^2*e^(6*d*x + 6*c) - 64*a^2
*e^(4*d*x + 4*c) + 94*a*b*e^(4*d*x + 4*c) - 30*b^2*e^(4*d*x + 4*c) - 14*a*b*e^(2*d*x + 2*c) + 20*b^2*e^(2*d*x
+ 2*c) + 4*a*b - 5*b^2)/((a^3 - a^2*b)*(b*e^(10*d*x + 10*c) - 5*b*e^(8*d*x + 8*c) - 16*a*e^(6*d*x + 6*c) + 10*
b*e^(6*d*x + 6*c) + 16*a*e^(4*d*x + 4*c) - 10*b*e^(4*d*x + 4*c) + 5*b*e^(2*d*x + 2*c) - b)*d)

________________________________________________________________________________________

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 )}^2} \,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)^2),x)

[Out]

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

________________________________________________________________________________________