3.1.56 \(\int \frac {\text {csch}(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [56]

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

[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 = 0.18, 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 = {3265, 425, 541, 536, 212, 211} \begin {gather*} -\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 ) \text {ArcTan}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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]*Cosh[c + d*x])/Sqrt[a - b]])/(a^3*(a - b)^(5/2)*d) - A
rcTanh[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 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 541

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 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 )^3} \, dx &=-\frac {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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] Result contains complex when optimal does not.
time = 2.23, size = 237, normalized size = 1.43 \begin {gather*} -\frac {\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\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 ) \text {ArcTan}\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+b \cosh (2 (c+d x)))^2}+\frac {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 \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^3 d} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(152)=304\).
time = 1.74, size = 308, normalized size = 1.86

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\)	\(308\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\)	\(308\)
risch	\(-\frac {{\mathrm e}^{d x +c} b \left (7 a b \,{\mathrm e}^{6 d x +6 c}-4 b^{2} {\mathrm e}^{6 d x +6 c}+36 a^{2} {\mathrm e}^{4 d x +4 c}-31 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}+36 a^{2} {\mathrm e}^{2 d x +2 c}-31 a b \,{\mathrm e}^{2 d x +2 c}+4 b^{2} {\mathrm e}^{2 d x +2 c}+7 a b -4 b^{2}\right )}{4 d \,a^{2} \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{3} d}+\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}-\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}+\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}\)	\(571\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(1/a^3*ln(tanh(1/2*d*x+1/2*c))-2/a^3*b*((-1/8*(9*a^2-28*a*b+16*b^2)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^
6+3/8*(9*a^3-30*a^2*b+40*a*b^2-16*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-1/8*a*(27*a^2-68*a*b+32*b^2)/(a^2
-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2+3/8*a^2*(3*a-2*b)/(a^2-2*a*b+b^2))/(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)^2+1/16*(15*a^2-20*a*b+8*b^2)/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4
*(2*a*tanh(1/2*d*x+1/2*c)^2-2*a+4*b)/(a*b-b^2)^(1/2))))

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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)/(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5242 vs. \(2 (152) = 304\).
time = 0.53, size = 9815, normalized size = 59.13 \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)^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 ...

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

[Out]

Timed out

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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)^3,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 )}^3} \,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)^3),x)

[Out]

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

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