3.1.40 \(\int \frac {\text {csch}^4(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [40]

Optimal. Leaf size=123 \[ -\frac {(3 a-2 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{7/2} d}+\frac {(a-b) \coth (c+d x)}{(a+b)^3 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^2 d}-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

(a-b)*coth(d*x+c)/(a+b)^3/d-1/3*coth(d*x+c)^3/(a+b)^2/d-1/2*(3*a-2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))
*b^(1/2)/(a+b)^(7/2)/d-1/2*a*b*tanh(d*x+c)/(a+b)^3/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]
time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4217, 467, 1275, 214} \begin {gather*} -\frac {\sqrt {b} (3 a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 d (a+b)^{7/2}}-\frac {a b \tanh (c+d x)}{2 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\coth ^3(c+d x)}{3 d (a+b)^2}+\frac {(a-b) \coth (c+d x)}{d (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

-1/2*((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/((a + b)^(7/2)*d) + ((a - b)*Coth[c +
d*x])/((a + b)^3*d) - Coth[c + d*x]^3/(3*(a + b)^2*d) - (a*b*Tanh[c + d*x])/(2*(a + b)^3*d*(a + b - b*Tanh[c +
d*x]^2))

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 467

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 4217

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1
+ ff^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rubi steps

\begin {align*} \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {b \text {Subst}\left (\int \frac {\frac {2}{b (a+b)}-\frac {2 a x^2}{b (a+b)^2}-\frac {a x^4}{(a+b)^3}}{x^4 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {b \text {Subst}\left (\int \left (\frac {2}{b (a+b)^2 x^4}-\frac {2 (a-b)}{b (a+b)^3 x^2}+\frac {-3 a+2 b}{(a+b)^3 \left (a+b-b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {(a-b) \coth (c+d x)}{(a+b)^3 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^2 d}-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {((3 a-2 b) b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^3 d}\\ &=-\frac {(3 a-2 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{7/2} d}+\frac {(a-b) \coth (c+d x)}{(a+b)^3 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^2 d}-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(295\) vs. \(2(123)=246\).
time = 3.90, size = 295, normalized size = 2.40 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (-2 (a+b) (a+2 b+a \cosh (2 (c+d x))) \coth (c) \text {csch}^2(c+d x)-\frac {3 (3 a-2 b) b \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}-4 (a-2 b) (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)+2 (a+b) (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}^3(c+d x) \sinh (d x)-3 a b \text {sech}(2 c) \sinh (2 d x)+3 b (a+2 b) \tanh (2 c)\right )}{24 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(-2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c]*Csch[c +
d*x]^2 - (3*(3*a - 2*b)*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x])
)/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(Sqr
t[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - 4*(a - 2*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Si
nh[d*x] + 2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh[d*x] - 3*a*b*Sech[2*c]*Sinh[2
*d*x] + 3*b*(a + 2*b)*Tanh[2*c]))/(24*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(323\) vs. \(2(109)=218\).
time = 2.62, size = 324, normalized size = 2.63

method	result	size
derivativedivides	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {2 b \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a -2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a +5 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(324\)
default	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {2 b \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a -2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a +5 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(324\)
risch	\(-\frac {9 a b \,{\mathrm e}^{8 d x +8 c}-6 b^{2} {\mathrm e}^{8 d x +8 c}+12 a^{2} {\mathrm e}^{6 d x +6 c}+18 a b \,{\mathrm e}^{6 d x +6 c}+66 b^{2} {\mathrm e}^{6 d x +6 c}+20 a^{2} {\mathrm e}^{4 d x +4 c}+44 a b \,{\mathrm e}^{4 d x +4 c}-66 b^{2} {\mathrm e}^{4 d x +4 c}+4 a^{2} {\mathrm e}^{2 d x +2 c}-18 a b \,{\mathrm e}^{2 d x +2 c}+38 b^{2} {\mathrm e}^{2 d x +2 c}-4 a^{2}+11 a b}{3 d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right ) a}{4 \left (a +b \right )^{4} d}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right ) b}{2 \left (a +b \right )^{4} d}-\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right ) a}{4 \left (a +b \right )^{4} d}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right ) b}{2 \left (a +b \right )^{4} d}\)	\(418\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8/(a^2+2*a*b+b^2)/(a+b)*(1/3*a*tanh(1/2*d*x+1/2*c)^3+1/3*b*tanh(1/2*d*x+1/2*c)^3-3*a*tanh(1/2*d*x+1/2*
c)+5*b*tanh(1/2*d*x+1/2*c))+2*b/(a+b)^3*((-1/2*a*tanh(1/2*d*x+1/2*c)^3-1/2*a*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*
d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)+1/2*(3*a-2*b)*(-
1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^
(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))))-1/24/(a+b)
^2/tanh(1/2*d*x+1/2*c)^3-1/8/(a+b)^3*(-3*a+5*b)/tanh(1/2*d*x+1/2*c))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (112) = 224\).
time = 0.57, size = 430, normalized size = 3.50 \begin {gather*} \frac {{\left (3 \, a b - 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {4 \, a^{2} - 11 \, a b - 2 \, {\left (2 \, a^{2} - 9 \, a b + 19 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (10 \, a^{2} + 22 \, a b - 33 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, {\left (2 \, a^{2} + 3 \, a b + 11 \, b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, {\left (3 \, a b - 2 \, b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{3 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{4} - a^{3} b - 9 \, a^{2} b^{2} - 11 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{4} + 9 \, a^{3} b + 21 \, a^{2} b^{2} + 19 \, a b^{3} + 6 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (a^{4} + 9 \, a^{3} b + 21 \, a^{2} b^{2} + 19 \, a b^{3} + 6 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} - a^{3} b - 9 \, a^{2} b^{2} - 11 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/4*(3*a*b - 2*b^2)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*s
qrt((a + b)*b)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/3*(4*a^2 - 11*a*b - 2*(2*a^2 - 9*a*b
+ 19*b^2)*e^(-2*d*x - 2*c) - 2*(10*a^2 + 22*a*b - 33*b^2)*e^(-4*d*x - 4*c) - 6*(2*a^2 + 3*a*b + 11*b^2)*e^(-6*
d*x - 6*c) - 3*(3*a*b - 2*b^2)*e^(-8*d*x - 8*c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 - (a^4 - a^3*b - 9*a^2*b^
2 - 11*a*b^3 - 4*b^4)*e^(-2*d*x - 2*c) - 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*e^(-4*d*x - 4*c) +
2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*e^(-6*d*x - 6*c) + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b
^4)*e^(-8*d*x - 8*c) - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*e^(-10*d*x - 10*c))*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2933 vs. \(2 (112) = 224\).
time = 0.43, size = 6143, normalized size = 49.94 \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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/12*(12*(3*a*b - 2*b^2)*cosh(d*x + c)^8 + 96*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(3*a*b - 2*
b^2)*sinh(d*x + c)^8 + 24*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^6 + 24*(14*(3*a*b - 2*b^2)*cosh(d*x + c)^2 +
2*a^2 + 3*a*b + 11*b^2)*sinh(d*x + c)^6 + 48*(14*(3*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(2*a^2 + 3*a*b + 11*b^2)*
cosh(d*x + c))*sinh(d*x + c)^5 + 8*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^4 + 8*(105*(3*a*b - 2*b^2)*cosh(d*
x + c)^4 + 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^2 + 10*a^2 + 22*a*b - 33*b^2)*sinh(d*x + c)^4 + 32*(21*(3
*a*b - 2*b^2)*cosh(d*x + c)^5 + 15*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^3 + (10*a^2 + 22*a*b - 33*b^2)*cosh(
d*x + c))*sinh(d*x + c)^3 + 8*(2*a^2 - 9*a*b + 19*b^2)*cosh(d*x + c)^2 + 8*(42*(3*a*b - 2*b^2)*cosh(d*x + c)^6
+ 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^4 + 6*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^2 + 2*a^2 - 9*a*b
+ 19*b^2)*sinh(d*x + c)^2 + 3*((3*a^2 - 2*a*b)*cosh(d*x + c)^10 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x +
c)^9 + (3*a^2 - 2*a*b)*sinh(d*x + c)^10 - (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^8 + (45*(3*a^2 - 2*a*b)*cosh(
d*x + c)^2 - 3*a^2 + 14*a*b - 8*b^2)*sinh(d*x + c)^8 + 8*(15*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 - (3*a^2 - 14*a*b
+ 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^6 + 2*(105*(3*a^2 - 2*a*b
)*cosh(d*x + c)^4 - 14*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^2 - 3*a^2 - 16*a*b + 12*b^2)*sinh(d*x + c)^6 + 4
*(63*(3*a^2 - 2*a*b)*cosh(d*x + c)^5 - 14*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 + 16*a*b - 12*b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(105*(3*a^2 - 2*a*b)*cosh(
d*x + c)^6 - 35*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^4 - 15*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^2 + 3*a^
2 + 16*a*b - 12*b^2)*sinh(d*x + c)^4 + 8*(15*(3*a^2 - 2*a*b)*cosh(d*x + c)^7 - 7*(3*a^2 - 14*a*b + 8*b^2)*cosh
(d*x + c)^5 - 5*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^3 + (3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x
+ c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^2 + (45*(3*a^2 - 2*a*b)*cosh(d*x + c)^8 - 28*(3*a^2 - 14*a*b +
8*b^2)*cosh(d*x + c)^6 - 30*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^4 + 12*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x
+ c)^2 + 3*a^2 - 14*a*b + 8*b^2)*sinh(d*x + c)^2 - 3*a^2 + 2*a*b + 2*(5*(3*a^2 - 2*a*b)*cosh(d*x + c)^9 - 4*(
3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^7 - 6*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 16*a*b - 12
*b^2)*cosh(d*x + c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d
*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3
*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*
a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)
+ (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*
sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x
+ c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - 16*a^2 + 44*a*b + 16*(6*(3*a*b
- 2*b^2)*cosh(d*x + c)^7 + 9*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^5 + 2*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x
+ c)^3 + (2*a^2 - 9*a*b + 19*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d
*x + c)^10 + 10*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + 3*a^3*b + 3*a^2*b
^2 + a*b^3)*d*sinh(d*x + c)^10 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^8 + (45*(a^4 + 3
*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d)*sinh(d*x + c)^
8 - 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a
*b^3)*d*cosh(d*x + c)^3 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(1
05*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 - 14*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*c
osh(d*x + c)^2 - (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^6 + 2*(a^4 + 9*a^3*b + 21*a^
2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^4 + 4*(63*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^5 - 14
*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^3 - 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6
*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^6 - 35*(a^
4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^4 - 15*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^
4)*d*cosh(d*x + c)^2 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^4 + (a^4 - a^3*b - 9*a
^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^2 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^7 - 7
*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^5 - 5*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6
*b^4)*d*cosh(d*x + c)^3 + (a^4 + 9*a^3*b + 21*a...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**4/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Integral(csch(c + d*x)**4/(a + b*sech(c + d*x)**2)**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (112) = 224\).
time = 0.63, size = 253, normalized size = 2.06 \begin {gather*} -\frac {\frac {3 \, {\left (3 \, a b - 2 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}} + \frac {8 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/6*(3*(3*a*b - 2*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*sqrt(-a*b - b^2)) - 6*(a*b*e^(2*d*x + 2*c) + 2*b^2*e^(2*d*x + 2*c) + a*b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^
3)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)) + 8*(3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*
x + 2*c) - 3*b*e^(2*d*x + 2*c) - a + 2*b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)^3))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^2),x)

[Out]

int(cosh(c + d*x)^4/(sinh(c + d*x)^4*(b + a*cosh(c + d*x)^2)^2), x)

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