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

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

[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))

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Rubi [A]  time = 1.1577, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3217, 1334, 1669, 1664, 1166, 208} \[ \frac{b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 d (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}+\frac{b \tanh (c+d x) \left (\frac{2 a^2 (9 a-17 b)}{(a-b)^3}-\frac{\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac{3 \sqrt{b} \left (-34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{3 \sqrt{b} \left (34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{\coth (c+d x)}{a^3 d} \]

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 3217

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]

Rule 1334

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 1669

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 1664

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 1166

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 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}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 3.45587, size = 357, normalized size = 0.99 \[ \frac{\frac{4 b \sinh (2 (c+d x)) \left (28 a^2+b (13 b-19 a) \cosh (2 (c+d x))+3 a b-13 b^2\right )}{(a-b)^2 (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+\frac{3 \sqrt{b} \left (34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\left (\sqrt{a}+\sqrt{b}\right )^2 \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{3 \sqrt{b} \left (-34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\left (\sqrt{a}-\sqrt{b}\right )^2 \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{128 a b \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{(a-b) (-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}-64 \coth (c+d x)}{64 a^3 d} \]

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]  time = 0.158, size = 2747, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-3/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-19/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*ta
nh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a^2*
b^3/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-19/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2
*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a^2*b^3/(a^2-2*a*b+b^2)*tanh(1/2*d*x
+1/2*c)^11-3/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*
x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^15+25/4/d/(tanh(1/2*d*x+1/
2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^
2*a+a)^2/a*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9+25/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a
+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a*b^2/(a^2-2*a*b+b^2)*tan
h(1/2*d*x+1/2*c)^7+153/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*t
anh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7*b^3-2/d/(tanh(1/
2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x
+1/2*c)^2*a+a)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3*b^2-2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c
)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a/(a^2-2*a*b+b^2)*ta
nh(1/2*d*x+1/2*c)^13*b^2+153/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-
16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9*b^3-7/2/d/
(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh
(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^5*b^2-7/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2
*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b
+b^2)/a*tanh(1/2*d*x+1/2*c)^11*b^2-3/64/d*b/a^3/(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))+17/4/d*b^3/a^2/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2
*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b
+b^2)*tanh(1/2*d*x+1/2*c)^13-52/d*b^4/a^3/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/
2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-52/d*
b^4/a^3/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^
4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9+17/4/d*b^3/a^2/(tanh(1/2*d*x+1/2*c)^8*a
-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2
/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-45/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*
x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^1
3-45/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)
^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+9/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tan
h(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-
2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^15*b+9/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/
2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)*b-1/2/d
/a^3*tanh(1/2*d*x+1/2*c)-1/2/d/a^3/tanh(1/2*d*x+1/2*c)+81/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6
*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(
1/2*d*x+1/2*c)^11+81/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tan
h(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-45/8/d/(tanh(1/2*d*x
+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*
c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-45/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+
6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2
*d*x+1/2*c)^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [A]  time = 23.5837, size = 660, normalized size = 1.84 \begin{align*} -\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}}{16 \,{\left (a^{5} d - 2 \, a^{4} b d + a^{3} b^{2} d\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{2}{a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}

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