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

Optimal. Leaf size=166 \[ -\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b (7 a-4 b) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 d (a-b)^{5/2}}-\frac {b \cosh (c+d x)}{4 a d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2} \]

[Out]

-arctanh(cosh(d*x+c))/a^3/d-1/4*b*cosh(d*x+c)/a/(a-b)/d/(a-b+b*cosh(d*x+c)^2)^2-1/8*(7*a-4*b)*b*cosh(d*x+c)/a^
2/(a-b)^2/d/(a-b+b*cosh(d*x+c)^2)-1/8*(15*a^2-20*a*b+8*b^2)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^
3/(a-b)^(5/2)/d

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Rubi [A]  time = 0.27, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3186, 414, 527, 522, 206, 205} \[ -\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 d (a-b)^{5/2}}-\frac {b (7 a-4 b) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x)}{4 a d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*a^3*(a - b)^(5/2)*d) - Arc
Tanh[Cosh[c + d*x]]/(a^3*d) - (b*Cosh[c + d*x])/(4*a*(a - b)*d*(a - b + b*Cosh[c + d*x]^2)^2) - ((7*a - 4*b)*b
*Cosh[c + d*x])/(8*a^2*(a - b)^2*d*(a - b + b*Cosh[c + d*x]^2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

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

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {-4 a+b+3 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {8 a^2-9 a b+4 b^2-(7 a-4 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 (a-b)^{5/2} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 3.37, size = 237, normalized size = 1.43 \[ -\frac {\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {8 a^2 b \cosh (c+d x)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)^2}+\frac {2 a b (7 a-4 b) \cosh (c+d x)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)}-8 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^
(5/2) + (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b
)^(5/2) + (8*a^2*b*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])^2) + (2*a*(7*a - 4*b)*b*Cosh[c + d*
x])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])) - 8*Log[Tanh[(c + d*x)/2]])/(a^3*d)

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fricas [B]  time = 0.89, size = 9815, normalized size = 59.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^7 + 28*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(
7*a^2*b^2 - 4*a*b^3)*sinh(d*x + c)^7 + 4*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^5 + 4*(36*a^3*b - 31*
a^2*b^2 + 4*a*b^3 + 21*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(7*a^2*b^2 - 4*a*b^3)*co
sh(d*x + c)^3 + (36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(36*a^3*b - 31*a^2*b^2 +
4*a*b^3)*cosh(d*x + c)^3 + 4*(35*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^4 + 36*a^3*b - 31*a^2*b^2 + 4*a*b^3 + 10*
(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)
^5 + 10*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^3 + 3*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^2 - ((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^8 + 8*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d
*x + c)*sinh(d*x + c)^7 + (15*a^2*b^2 - 20*a*b^3 + 8*b^4)*sinh(d*x + c)^8 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^
3 - 8*b^4)*cosh(d*x + c)^6 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4 + 7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^3 + 3*(30*a^3*b - 55*a^2*
b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24
*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 120*a^4 - 280*a^3*b + 269*a^2*
b^2 - 124*a*b^3 + 24*b^4 + 30*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15
*a^2*b^2 - 20*a*b^3 + 8*b^4 + 8*(7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^5 + 10*(30*a^3*b - 55*a^2*b^2
+ 36*a*b^3 - 8*b^4)*cosh(d*x + c)^3 + (120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^2 + 4*(7*(15*a^2*b^2 - 20*a*b^3
+ 8*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^4 + 30*a^3*b - 55*a^2*b
^2 + 36*a*b^3 - 8*b^4 + 3*(120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^7 + 3*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh
(d*x + c)^5 + (120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c)^3 + (30*a^3*b - 55*a^2*b^
2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a - b))*log((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 - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*si
nh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a - b)*cosh(d*x + c)^3 +
3*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a - b)*sinh(d*x + c)^3 + (a - b)*cosh(d*x + c) + (3*(a - b)*cosh(d
*x + c)^2 + a - b)*sinh(d*x + c))*sqrt(-b/(a - b)) + 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)) + 4*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c) + 16*((
a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^8 + 8*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b^
2 - 2*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(2*a^3*b - 5*
a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b^2 - 2*a*b
^3 + b^4)*cosh(d*x + c)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4
- 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^4
+ 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(2*a^3*b - 5*
a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c))*s
inh(d*x + c)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh
(d*x + c)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 +
3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b^2 - 2*a*b^3
+ b^4)*cosh(d*x + c)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^4 - 24*a^3*b + 27*a^2
*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))
*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 16*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^8 + 8*(a^2*b^2 - 2*a*b^3
+ b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4
*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x
+ c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 -
b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2
*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*
b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*
a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^4 - 24*a^3*b +
27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d
*x + c)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x
+ c)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^
4)*cosh(d*x + c)^5 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a^3*b - 5*a^2*b^2
+ 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(7*(7*a^2*b^2 - 4*a
*b^3)*cosh(d*x + c)^6 + 5*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^4 + 7*a^2*b^2 - 4*a*b^3 + 3*(36*a^3*
b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 +
8*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*sinh(d*
x + c)^8 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b
^4)*d*cosh(d*x + c)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(8*a^7 - 24*a^6*b +
27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)
^3 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^2 - 2*a^4*b
^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 30*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + (8*a^7 -
24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*
b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^5 + 10*(2*a^6*b - 5*a^5*b^2 + 4*
a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c
))*sinh(d*x + c)^3 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^6 + 15*(2*a^6*b - 5*a^5*b^2 + 4*a^4*
b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^
2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d + 8*((a
^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^7 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c
)^5 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^3 + (2*a^6*b - 5*a^5*b^2 + 4*a^
4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^7 + 14*(7*a^2*b^
2 - 4*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(7*a^2*b^2 - 4*a*b^3)*sinh(d*x + c)^7 + 2*(36*a^3*b - 31*a^2*b^
2 + 4*a*b^3)*cosh(d*x + c)^5 + 2*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3 + 21*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c)^5 + 10*(7*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 + (36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c
))*sinh(d*x + c)^4 + 2*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^3 + 2*(35*(7*a^2*b^2 - 4*a*b^3)*cosh(d*
x + c)^4 + 36*a^3*b - 31*a^2*b^2 + 4*a*b^3 + 10*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^3 + 2*(21*(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^5 + 10*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^3 + 3*
(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x +
c)^8 + 8*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2*b^2 - 20*a*b^3 + 8*b^4)*sinh
(d*x + c)^8 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^6 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b
^3 - 8*b^4 + 7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 20*a*b^3
+ 8*b^4)*cosh(d*x + c)^3 + 3*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12
0*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*
cosh(d*x + c)^4 + 120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4 + 30*(30*a^3*b - 55*a^2*b^2 + 36*a*b^
3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^2*b^2 - 20*a*b^3 + 8*b^4 + 8*(7*(15*a^2*b^2 - 20*a*b^3 + 8*
b^4)*cosh(d*x + c)^5 + 10*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^3 + (120*a^4 - 280*a^3*b +
269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4
)*cosh(d*x + c)^2 + 4*(7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b - 55*a^2*b^2 + 36*a*b^
3 - 8*b^4)*cosh(d*x + c)^4 + 30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4 + 3*(120*a^4 - 280*a^3*b + 269*a^2*b^2 -
124*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^7 + 3
*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^5 + (120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 +
24*b^4)*cosh(d*x + c)^3 + (30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a
- b))*arctan(1/2*sqrt(b/(a - b))*(cosh(d*x + c) + sinh(d*x + c))) - ((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x
+ c)^8 + 8*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2*b^2 - 20*a*b^3 + 8*b^4)*sin
h(d*x + c)^8 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^6 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*
b^3 - 8*b^4 + 7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 20*a*b^3
+ 8*b^4)*cosh(d*x + c)^3 + 3*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(1
20*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)
*cosh(d*x + c)^4 + 120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3 + 24*b^4 + 30*(30*a^3*b - 55*a^2*b^2 + 36*a*b
^3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^2*b^2 - 20*a*b^3 + 8*b^4 + 8*(7*(15*a^2*b^2 - 20*a*b^3 + 8
*b^4)*cosh(d*x + c)^5 + 10*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^3 + (120*a^4 - 280*a^3*b +
269*a^2*b^2 - 124*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^
4)*cosh(d*x + c)^2 + 4*(7*(15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b - 55*a^2*b^2 + 36*a*b
^3 - 8*b^4)*cosh(d*x + c)^4 + 30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4 + 3*(120*a^4 - 280*a^3*b + 269*a^2*b^2
- 124*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2 - 20*a*b^3 + 8*b^4)*cosh(d*x + c)^7 +
3*(30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c)^5 + (120*a^4 - 280*a^3*b + 269*a^2*b^2 - 124*a*b^3
+ 24*b^4)*cosh(d*x + c)^3 + (30*a^3*b - 55*a^2*b^2 + 36*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a
- b))*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - 3*b)*cos
h(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))*sqrt(b/(a - b))/b) + 2*(7*a^2*b^2 - 4*a*b^3)*cos
h(d*x + c) + 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^8 + 8*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x
+ c)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^6
+ 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7
*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)
*cosh(d*x + c)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)
*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^5 +
10*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)
*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*(a^2*b^2 - 2*
a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^4 + 2*a^3*b - 5*a^2*b^2
+ 4*a*b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((
a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^4 -
24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c
))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^8 + 8*(a
^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(2*a^3*b
- 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3
+ b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(2*a^3*b - 5*a^2*
b^2 + 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cos
h(d*x + c)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*
b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*
(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a
^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b
^3 - b^4)*cosh(d*x + c)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3
- b^4)*cosh(d*x + c)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 +
3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(d*x + c)^7 + 3*(2*a^3*b - 5*a^2*b^
2 + 4*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a
^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*
(7*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^6 + 5*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^4 + 7*a^2*b^2 - 4*a*
b^3 + 3*(36*a^3*b - 31*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*c
osh(d*x + c)^8 + 8*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^2 - 2*a^4*b^3 + a^
3*b^4)*d*sinh(d*x + c)^8 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^2 - 2
*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(8*
a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)
*d*cosh(d*x + c)^3 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a
^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 30*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x +
c)^2 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b - 5*a^5*b^2 +
4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^5 + 10*(2*a^6*b
- 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4
)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^6 + 15*(2*a^6*b - 5*
a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*
d*cosh(d*x + c)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 + (a^5*b^2 - 2*a^4*b^3 + a^
3*b^4)*d + 8*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^7 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4
)*d*cosh(d*x + c)^5 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^3 + (2*a^6*b -
5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-85,-18]Warning, need to choose a branch for the root of a polynomial with parameters. Th
is might be wrong.The choice was done assuming [a,b]=[33,-80]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-98,-18]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[-57,-10]Warning, need to choose a branch for the root of a polynomial with parameters. This might be
wrong.The choice was done assuming [a,b]=[-57,-3]Warning, need to choose a branch for the root of a polynomial
with parameters. This might be wrong.The choice was done assuming [a,b]=[-53,60]Undef/Unsigned Inf encountere
d in limitEvaluation time: 1Limit: Max order reached or unable to make series expansion Error: Bad Argument Va
lue

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maple [B]  time = 0.13, size = 1145, normalized size = 6.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^6*b-7/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a/(a^2
-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*b^2+4/d/a^2*b^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/
2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-27/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2
*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+45/2/d/(tanh(1/2*d*x+1/2*c)^4*a
-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-30/d/(ta
nh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*
x+1/2*c)^4*b^3+12/d/a^3*b^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/
(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+27/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x
+1/2*c)^2*b+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-17/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2
*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2*b^2+8/d/a^2*b^3/(tanh(1/2*d*x+1/2*c)
^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-9/4/d/(tan
h(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*b+3/2/d/a*b^2/(t
anh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)-15/8/d/a*b/(a^
2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))+5/2/d/a^2*b^2/(a^
2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))-1/d/a^3*b^3/(a^2-
2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))+1/d/a^3*ln(tanh(1/2
*d*x+1/2*c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (7 \, a b^{2} e^{\left (7 \, c\right )} - 4 \, b^{3} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (36 \, a^{2} b e^{\left (5 \, c\right )} - 31 \, a b^{2} e^{\left (5 \, c\right )} + 4 \, b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (36 \, a^{2} b e^{\left (3 \, c\right )} - 31 \, a b^{2} e^{\left (3 \, c\right )} + 4 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (7 \, a b^{2} e^{c} - 4 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{4} b^{2} d - 2 \, a^{3} b^{3} d + a^{2} b^{4} d + {\left (a^{4} b^{2} d e^{\left (8 \, c\right )} - 2 \, a^{3} b^{3} d e^{\left (8 \, c\right )} + a^{2} b^{4} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (2 \, a^{5} b d e^{\left (6 \, c\right )} - 5 \, a^{4} b^{2} d e^{\left (6 \, c\right )} + 4 \, a^{3} b^{3} d e^{\left (6 \, c\right )} - a^{2} b^{4} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (8 \, a^{6} d e^{\left (4 \, c\right )} - 24 \, a^{5} b d e^{\left (4 \, c\right )} + 27 \, a^{4} b^{2} d e^{\left (4 \, c\right )} - 14 \, a^{3} b^{3} d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (2 \, a^{5} b d e^{\left (2 \, c\right )} - 5 \, a^{4} b^{2} d e^{\left (2 \, c\right )} + 4 \, a^{3} b^{3} d e^{\left (2 \, c\right )} - a^{2} b^{4} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} - \frac {\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a^{3} d} + \frac {\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a^{3} d} - 2 \, \int \frac {{\left (15 \, a^{2} b e^{\left (3 \, c\right )} - 20 \, a b^{2} e^{\left (3 \, c\right )} + 8 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (15 \, a^{2} b e^{c} - 20 \, a b^{2} e^{c} + 8 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} + {\left (a^{5} b e^{\left (4 \, c\right )} - 2 \, a^{4} b^{2} e^{\left (4 \, c\right )} + a^{3} b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (2 \, a^{6} e^{\left (2 \, c\right )} - 5 \, a^{5} b e^{\left (2 \, c\right )} + 4 \, a^{4} b^{2} e^{\left (2 \, c\right )} - a^{3} b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((7*a*b^2*e^(7*c) - 4*b^3*e^(7*c))*e^(7*d*x) + (36*a^2*b*e^(5*c) - 31*a*b^2*e^(5*c) + 4*b^3*e^(5*c))*e^(5
*d*x) + (36*a^2*b*e^(3*c) - 31*a*b^2*e^(3*c) + 4*b^3*e^(3*c))*e^(3*d*x) + (7*a*b^2*e^c - 4*b^3*e^c)*e^(d*x))/(
a^4*b^2*d - 2*a^3*b^3*d + a^2*b^4*d + (a^4*b^2*d*e^(8*c) - 2*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x)
+ 4*(2*a^5*b*d*e^(6*c) - 5*a^4*b^2*d*e^(6*c) + 4*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^6*d
*e^(4*c) - 24*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) - 14*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) +
4*(2*a^5*b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) + 4*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) - log((e^(d*
x + c) + 1)*e^(-c))/(a^3*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) - 2*integrate(1/8*((15*a^2*b*e^(3*c) - 20*
a*b^2*e^(3*c) + 8*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c - 20*a*b^2*e^c + 8*b^3*e^c)*e^(d*x))/(a^5*b - 2*a^4*b
^2 + a^3*b^3 + (a^5*b*e^(4*c) - 2*a^4*b^2*e^(4*c) + a^3*b^3*e^(4*c))*e^(4*d*x) + 2*(2*a^6*e^(2*c) - 5*a^5*b*e^
(2*c) + 4*a^4*b^2*e^(2*c) - a^3*b^3*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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