3.1.45 \(\int \frac {\text {csch}(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [45]

Optimal. Leaf size=156 \[ -\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{5/2} d}+\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )} \]

[Out]

-arctanh(cosh(d*x+c))/a^3/d+1/4*b*sech(d*x+c)/a/(a+b)/d/(a+b-b*sech(d*x+c)^2)^2+1/8*b*(7*a+4*b)*sech(d*x+c)/a^
2/(a+b)^2/d/(a+b-b*sech(d*x+c)^2)+1/8*(15*a^2+20*a*b+8*b^2)*arctanh(sech(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 = 0.19, antiderivative size = 156, 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 = {3745, 425, 541, 536, 213, 214} \begin {gather*} -\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 d (a+b)^2 \left (a-b \text {sech}^2(c+d x)+b\right )}+\frac {\sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^3 d (a+b)^{5/2}}+\frac {b \text {sech}(c+d x)}{4 a d (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 213

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

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 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 3745

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m
+ 1)), x], x, Sec[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 \tanh ^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,\text {sech}(c+d x)\right )}{d}\\ &=\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^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,\text {sech}(c+d x)\right )}{4 a (a+b) d}\\ &=\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2+9 a b+4 b^2+b (7 a+4 b) x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(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,\text {sech}(c+d x)\right )}{8 a^3 (a+b)^2 d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{5/2} d}+\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.05, size = 236, normalized size = 1.51 \begin {gather*} \frac {\frac {i \sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{5/2}}+\frac {i \sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{5/2}}+\frac {8 a^2 b^2 \cosh (c+d x)}{(a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {2 a b (9 a+4 b) \cosh (c+d x)}{(a+b)^2 (a-b+(a+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*Tanh[c + d*x]^2)^3,x]

[Out]

((I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^
(5/2) + (I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(
a + b)^(5/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*a*b*(9*a + 4*b
)*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 8*Log[Tanh[(c + d*x)/2]])/(8*a^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(142)=284\).
time = 3.20, size = 304, normalized size = 1.95

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \left (9 a^{3}+30 a^{2} b +40 a \,b^{2}+16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}+68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a +2 b \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (15 a^{2}+20 a b +8 b^{2}\right ) \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\)	\(304\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \left (9 a^{3}+30 a^{2} b +40 a \,b^{2}+16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}+68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a +2 b \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (15 a^{2}+20 a b +8 b^{2}\right ) \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\)	\(304\)
risch	\(\frac {\left (9 a^{2} {\mathrm e}^{6 d x +6 c}+13 a b \,{\mathrm e}^{6 d x +6 c}+4 b^{2} {\mathrm e}^{6 d x +6 c}+27 a^{2} {\mathrm e}^{4 d x +4 c}+11 a b \,{\mathrm e}^{4 d x +4 c}-4 b^{2} {\mathrm e}^{4 d x +4 c}+27 a^{2} {\mathrm e}^{2 d x +2 c}+11 a b \,{\mathrm e}^{2 d x +2 c}-4 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+13 a b +4 b^{2}\right ) b \,{\mathrm e}^{d x +c}}{4 \left (a^{2}+2 a b +b^{2}\right ) \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} a^{2} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{3} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{3} d}+\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{3} d a}+\frac {5 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{3} d \,a^{2}}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{3} d a}-\frac {5 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{3} d \,a^{2}}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{3} d \,a^{3}}\)	\(572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)/(a+b*tanh(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)*arctanh(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*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((9*a^2*b*e^(7*c) + 13*a*b^2*e^(7*c) + 4*b^3*e^(7*c))*e^(7*d*x) + (27*a^2*b*e^(5*c) + 11*a*b^2*e^(5*c) - 4
*b^3*e^(5*c))*e^(5*d*x) + (27*a^2*b*e^(3*c) + 11*a*b^2*e^(3*c) - 4*b^3*e^(3*c))*e^(3*d*x) + (9*a^2*b*e^c + 13*
a*b^2*e^c + 4*b^3*e^c)*e^(d*x))/(a^6*d + 4*a^5*b*d + 6*a^4*b^2*d + 4*a^3*b^3*d + a^2*b^4*d + (a^6*d*e^(8*c) +
4*a^5*b*d*e^(8*c) + 6*a^4*b^2*d*e^(8*c) + 4*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x) + 4*(a^6*d*e^(6*c
) + 2*a^5*b*d*e^(6*c) - 2*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(3*a^6*d*e^(4*c) + 4*a^5*b*d*e^
(4*c) + 2*a^4*b^2*d*e^(4*c) + 4*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^6*d*e^(2*c) + 2*a^5*
b*d*e^(2*c) - 2*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) + lo
g((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^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 + (a^6*e^(4
*c) + 3*a^5*b*e^(4*c) + 3*a^4*b^2*e^(4*c) + a^3*b^3*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + a^5*b*e^(2*c) - 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. 5783 vs. \(2 (148) = 296\).
time = 0.56, size = 10716, normalized size = 68.69 \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*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^7 + 28*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)*
sinh(d*x + c)^6 + 4*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*sinh(d*x + c)^7 + 4*(27*a^3*b + 11*a^2*b^2 - 4*a*b^3)*cos
h(d*x + c)^5 + 4*(27*a^3*b + 11*a^2*b^2 - 4*a*b^3 + 21*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^5 + 20*(7*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^3 + (27*a^3*b + 11*a^2*b^2 - 4*a*b^3)*cosh(d
*x + c))*sinh(d*x + c)^4 + 4*(27*a^3*b + 11*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 + 4*(35*(9*a^3*b + 13*a^2*b^2 +
4*a*b^3)*cosh(d*x + c)^4 + 27*a^3*b + 11*a^2*b^2 - 4*a*b^3 + 10*(27*a^3*b + 11*a^2*b^2 - 4*a*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^3 + 4*(21*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^5 + 10*(27*a^3*b + 11*a^2*b^2 - 4
*a*b^3)*cosh(d*x + c)^3 + 3*(27*a^3*b + 11*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^4 + 50*a
^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^8 + 8*(15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*
cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*sinh(d*x + c)^8 + 4*(15*a^
4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4)*cosh(d*x + c)^6 + 4*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8
*b^4 + 7*(15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 +
50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^3 + 3*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b
^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 + 30*a^3*b + 29*a^2*b^2 + 44*a*b^3 + 24*b^4)*cosh(d*x + c)^4 +
2*(35*(15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 45*a^4 + 30*a^3*b + 29*a^2*b^2 + 4
4*a*b^3 + 24*b^4 + 30*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15
*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4 + 8*(7*(15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cos
h(d*x + c)^5 + 10*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4)*cosh(d*x + c)^3 + (45*a^4 + 30*a^3*b + 29
*a^2*b^2 + 44*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8
*b^4)*cosh(d*x + c)^2 + 4*(7*(15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^6 + 15*(15*a^4
+ 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4)*cosh(d*x + c)^4 + 15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4
+ 3*(45*a^4 + 30*a^3*b + 29*a^2*b^2 + 44*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 50*a^
3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4)*cosh(d*x + c)^7 + 3*(15*a^4 + 20*a^3*b - 7*a^2*b^2 - 20*a*b^3 - 8*b^4)*co
sh(d*x + c)^5 + (45*a^4 + 30*a^3*b + 29*a^2*b^2 + 44*a*b^3 + 24*b^4)*cosh(d*x + c)^3 + (15*a^4 + 20*a^3*b - 7*
a^2*b^2 - 20*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*log(((a + b)*cosh(d*x + c)^4 + 4*(a
+ b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh
(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (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)) + a + b)/((a + b)*cosh(d*x + c)
^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a +
b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x +
c) + a + b)) + 4*(9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cosh(d*x + c) - 16*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b
^4)*cosh(d*x + c)^8 + 8*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 4*a
^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(a^4
+ 2*a^3*b - 2*a*b^3 - b^4 + 7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 +
8*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + 4*a^3
*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^4 + 3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4 + 30*(a^4 + 2*
a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 + 8*(7*(a^
4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^3
+ (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b - 2*a*b^3
- b^4)*cosh(d*x + c)^2 + 4*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*b
- 2*a*b^3 - b^4)*cosh(d*x + c)^4 + a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 3*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 +
3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^7 + 3*(
a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^5 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)
^3 + (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(c + d*x)/(a + b*tanh(c + d*x)**2)**3, x)

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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*tanh(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 {tanh}\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*tanh(c + d*x)^2)^3),x)

[Out]

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

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