3.1.47 \(\int \frac {\text {csch}(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [47]
Optimal. Leaf size=110 \[ -\frac {(3 a-2 b) \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 a^2 (a-b)^{3/2} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x)}{2 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )} \]
[Out]
-arctanh(cosh(d*x+c))/a^2/d-1/2*b*cosh(d*x+c)/a/(a-b)/d/(a-b+b*cosh(d*x+c)^2)-1/2*(3*a-2*b)*arctan(cosh(d*x+c)
*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^2/(a-b)^(3/2)/d
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Rubi [A]
time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3265, 425, 536,
212, 211} \begin {gather*} -\frac {\sqrt {b} (3 a-2 b) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 a^2 d (a-b)^{3/2}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x)}{2 a d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
-1/2*((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a^2*(a - b)^(3/2)*d) - ArcTanh[Cosh[c
+ d*x]]/(a^2*d) - (b*Cosh[c + d*x])/(2*a*(a - b)*d*(a - b + b*Cosh[c + d*x]^2))
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 425
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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 536
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 3265
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 )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh (c+d x)}{2 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-2 a+b+b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{2 a (a-b) d}\\ &=-\frac {b \cosh (c+d x)}{2 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac {((3 a-2 b) b) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^2 (a-b) d}\\ &=-\frac {(3 a-2 b) \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 a^2 (a-b)^{3/2} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x)}{2 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.45, size = 176, normalized size = 1.60 \begin {gather*} \frac {\frac {\sqrt {b} (-3 a+2 b) \text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {\sqrt {b} (-3 a+2 b) \text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}-\frac {2 a b \cosh (c+d x)}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}+2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
((Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(3/2) + (Sqrt[b]*(
-3*a + 2*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(3/2) - (2*a*b*Cosh[c + d*x])
/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])) + 2*Log[Tanh[(c + d*x)/2]])/(2*a^2*d)
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Maple [A]
time = 1.50, size = 171, normalized size = 1.55
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a -b \right )}+\frac {a}{4 a -4 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a -2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a -b \right ) \sqrt {a b -b^{2}}}\right )}{a^{2}}}{d}\) |
\(171\) |
| default |
\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a -b \right )}+\frac {a}{4 a -4 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a -2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a -b \right ) \sqrt {a b -b^{2}}}\right )}{a^{2}}}{d}\) |
\(171\) |
| risch |
\(-\frac {b \,{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{4 \left (a -b \right )^{2} d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}-\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{4 \left (a -b \right )^{2} d a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}\) |
\(341\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
[Out]
1/d*(1/a^2*ln(tanh(1/2*d*x+1/2*c))-4*b/a^2*((-1/4*(-2*b+a)/(a-b)*tanh(1/2*d*x+1/2*c)^2+1/4*a/(a-b))/(a*tanh(1/
2*d*x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/8*(3*a-2*b)/(a-b)/(a*b-b^2)^(1/2)*arct
an(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2-2*a+4*b)/(a*b-b^2)^(1/2))))
________________________________________________________________________________________
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)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(4*c) - a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(2
*a^3*d*e^(2*c) - 3*a^2*b*d*e^(2*c) + a*b^2*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) + log
((e^(d*x + c) - 1)*e^(-c))/(a^2*d) - 2*integrate(1/2*((3*a*b*e^(3*c) - 2*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c -
2*b^2*e^c)*e^(d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(4*c) - a^2*b^2*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 3*a^3
*b*e^(2*c) + a^2*b^2*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. 1256 vs.
\(2 (98) = 196\).
time = 0.51, size = 2529, normalized size = 22.99 \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)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[-1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b*sinh(d*x + c)^3 + 4*a*b*cosh(d*x +
c) - ((3*a*b - 2*b^2)*cosh(d*x + c)^4 + 4*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - 2*b^2)*sin
h(d*x + c)^4 + 2*(6*a^2 - 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - 2*b^2)*cosh(d*x + c)^2 + 6*a^2 - 7*a*
b + 2*b^2)*sinh(d*x + c)^2 + 3*a*b - 2*b^2 + 4*((3*a*b - 2*b^2)*cosh(d*x + c)^3 + (6*a^2 - 7*a*b + 2*b^2)*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)*sinh(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)*s
inh(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*((a*b - b^2)*cosh(d*x + c)^4 + 4*(a*b - b^2)*cosh(d*x + c)*sinh(d*x +
c)^3 + (a*b - b^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a*b - b^2)*cosh(d*x + c)
^2 + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*b + b^2)
*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 4*((a*b - b^2)*cosh(d*x + c)^4 + 4*(a*
b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b - b^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2
+ 2*(3*(a*b - b^2)*cosh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(d
*x + c)^3 + (2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(3*
a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/((a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^3*b - a^2*b^2)*d*cosh(d*
x + c)*sinh(d*x + c)^3 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^4 + 2*(2*a^4 - 3*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2
+ 2*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + (2*a^4 - 3*a^3*b + a^2*b^2)*d)*sinh(d*x + c)^2 + (a^3*b - a^2*b^
2)*d + 4*((a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 + (2*a^4 - 3*a^3*b + a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), -
1/2*(2*a*b*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*b*sinh(d*x + c)^3 + 2*a*b*cosh(d*x + c)
+ ((3*a*b - 2*b^2)*cosh(d*x + c)^4 + 4*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - 2*b^2)*sinh(d
*x + c)^4 + 2*(6*a^2 - 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - 2*b^2)*cosh(d*x + c)^2 + 6*a^2 - 7*a*b +
2*b^2)*sinh(d*x + c)^2 + 3*a*b - 2*b^2 + 4*((3*a*b - 2*b^2)*cosh(d*x + c)^3 + (6*a^2 - 7*a*b + 2*b^2)*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))) - ((3*a*b -
2*b^2)*cosh(d*x + c)^4 + 4*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - 2*b^2)*sinh(d*x + c)^4 +
2*(6*a^2 - 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - 2*b^2)*cosh(d*x + c)^2 + 6*a^2 - 7*a*b + 2*b^2)*sinh
(d*x + c)^2 + 3*a*b - 2*b^2 + 4*((3*a*b - 2*b^2)*cosh(d*x + c)^3 + (6*a^2 - 7*a*b + 2*b^2)*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)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))*sqrt(b/(a - b))/b) + 2*((a*b
- b^2)*cosh(d*x + c)^4 + 4*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b - b^2)*sinh(d*x + c)^4 + 2*(2*a^2
- 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a*b - b^2)*cosh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + a*
b - b^2 + 4*((a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x +
c) + sinh(d*x + c) + 1) - 2*((a*b - b^2)*cosh(d*x + c)^4 + 4*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b
- b^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a*b - b^2)*cosh(d*x + c)^2 + 2*a^2 -
3*a*b + b^2)*sinh(d*x + c)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*b + b^2)*cosh(d*x + c
))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/((a^
3*b - a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b - a^2*b^2)*d*s
inh(d*x + c)^4 + 2*(2*a^4 - 3*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 +
(2*a^4 - 3*a^3*b + a^2*b^2)*d)*sinh(d*x + c)^2 + (a^3*b - a^2*b^2)*d + 4*((a^3*b - a^2*b^2)*d*cosh(d*x + c)^3
+ (2*a^4 - 3*a^3*b + a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))]
________________________________________________________________________________________
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)/(a+b*sinh(d*x+c)**2)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sinh(c + d*x)*(a + b*sinh(c + d*x)^2)^2),x)
[Out]
int(1/(sinh(c + d*x)*(a + b*sinh(c + d*x)^2)^2), x)
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