3.1.46 \(\int \frac {\text {csch}^2(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [46]

Optimal. Leaf size=126 \[ \frac {15 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} d}-\frac {15 \coth (c+d x)}{8 (a+b)^3 d}+\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \coth (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 126, 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, 296, 331, 214} \begin {gather*} \frac {15 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{7/2}}-\frac {15 \coth (c+d x)}{8 d (a+b)^3}+\frac {5 \coth (c+d x)}{8 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {\coth (c+d x)}{4 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(7/2)*d) - (15*Coth[c + d*x])/(8*(a + b)^
3*d) + Coth[c + d*x]/(4*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (5*Coth[c + d*x])/(8*(a + b)^2*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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \coth (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {15 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d}\\ &=-\frac {15 \coth (c+d x)}{8 (a+b)^3 d}+\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \coth (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {(15 b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^3 d}\\ &=\frac {15 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} d}-\frac {15 \coth (c+d x)}{8 (a+b)^3 d}+\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \coth (c+d x)}{8 (a+b)^2 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. \(741\) vs. \(2(126)=252\).
time = 5.35, size = 741, normalized size = 5.88 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (\frac {120 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)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\text {csch}(c) \text {csch}(c+d x) \text {sech}(2 c) \left (\left (-32 a^4-64 a^3 b+22 a^2 b^2+80 a b^3+16 b^4\right ) \sinh (d x)+2 a \left (16 a^3+23 a^2 b-27 a b^2-4 b^3\right ) \sinh (3 d x)-48 a^4 \sinh (2 c-d x)-128 a^3 b \sinh (2 c-d x)-106 a^2 b^2 \sinh (2 c-d x)+80 a b^3 \sinh (2 c-d x)+16 b^4 \sinh (2 c-d x)+48 a^4 \sinh (2 c+d x)+146 a^3 b \sinh (2 c+d x)+182 a^2 b^2 \sinh (2 c+d x)+80 a b^3 \sinh (2 c+d x)+16 b^4 \sinh (2 c+d x)-32 a^4 \sinh (4 c+d x)-82 a^3 b \sinh (4 c+d x)-54 a^2 b^2 \sinh (4 c+d x)-80 a b^3 \sinh (4 c+d x)-16 b^4 \sinh (4 c+d x)-8 a^4 \sinh (2 c+3 d x)+18 a^3 b \sinh (2 c+3 d x)+54 a^2 b^2 \sinh (2 c+3 d x)+8 a b^3 \sinh (2 c+3 d x)+32 a^4 \sinh (4 c+3 d x)+73 a^3 b \sinh (4 c+3 d x)+24 a^2 b^2 \sinh (4 c+3 d x)+8 a b^3 \sinh (4 c+3 d x)-8 a^4 \sinh (6 c+3 d x)-9 a^3 b \sinh (6 c+3 d x)-24 a^2 b^2 \sinh (6 c+3 d x)-8 a b^3 \sinh (6 c+3 d x)+8 a^4 \sinh (2 c+5 d x)-9 a^3 b \sinh (2 c+5 d x)-2 a^2 b^2 \sinh (2 c+5 d x)+9 a^3 b \sinh (4 c+5 d x)+2 a^2 b^2 \sinh (4 c+5 d x)+8 a^4 \sinh (6 c+5 d x)\right )}{a^2}\right )}{512 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((120*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)]
)^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Csch[c]*Csch[c + d*x]*Sech[2*c]*((
-32*a^4 - 64*a^3*b + 22*a^2*b^2 + 80*a*b^3 + 16*b^4)*Sinh[d*x] + 2*a*(16*a^3 + 23*a^2*b - 27*a*b^2 - 4*b^3)*Si
nh[3*d*x] - 48*a^4*Sinh[2*c - d*x] - 128*a^3*b*Sinh[2*c - d*x] - 106*a^2*b^2*Sinh[2*c - d*x] + 80*a*b^3*Sinh[2
*c - d*x] + 16*b^4*Sinh[2*c - d*x] + 48*a^4*Sinh[2*c + d*x] + 146*a^3*b*Sinh[2*c + d*x] + 182*a^2*b^2*Sinh[2*c
+ d*x] + 80*a*b^3*Sinh[2*c + d*x] + 16*b^4*Sinh[2*c + d*x] - 32*a^4*Sinh[4*c + d*x] - 82*a^3*b*Sinh[4*c + d*x
] - 54*a^2*b^2*Sinh[4*c + d*x] - 80*a*b^3*Sinh[4*c + d*x] - 16*b^4*Sinh[4*c + d*x] - 8*a^4*Sinh[2*c + 3*d*x] +
18*a^3*b*Sinh[2*c + 3*d*x] + 54*a^2*b^2*Sinh[2*c + 3*d*x] + 8*a*b^3*Sinh[2*c + 3*d*x] + 32*a^4*Sinh[4*c + 3*d
*x] + 73*a^3*b*Sinh[4*c + 3*d*x] + 24*a^2*b^2*Sinh[4*c + 3*d*x] + 8*a*b^3*Sinh[4*c + 3*d*x] - 8*a^4*Sinh[6*c +
3*d*x] - 9*a^3*b*Sinh[6*c + 3*d*x] - 24*a^2*b^2*Sinh[6*c + 3*d*x] - 8*a*b^3*Sinh[6*c + 3*d*x] + 8*a^4*Sinh[2*
c + 5*d*x] - 9*a^3*b*Sinh[2*c + 5*d*x] - 2*a^2*b^2*Sinh[2*c + 5*d*x] + 9*a^3*b*Sinh[4*c + 5*d*x] + 2*a^2*b^2*S
inh[4*c + 5*d*x] + 8*a^4*Sinh[6*c + 5*d*x]))/a^2))/(512*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(110)=220\).
time = 2.42, size = 299, normalized size = 2.37

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\frac {\left (-\frac {9 a}{8}-\frac {9 b}{8}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {27 a}{8}+\frac {b}{8}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {27 a}{8}+\frac {b}{8}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {9 a}{8}-\frac {9 b}{8}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {15 \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{32 \sqrt {b}\, \sqrt {a +b}}+\frac {15 \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{32 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{3}}}{d}\)	\(299\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\frac {\left (-\frac {9 a}{8}-\frac {9 b}{8}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {27 a}{8}+\frac {b}{8}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {27 a}{8}+\frac {b}{8}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {9 a}{8}-\frac {9 b}{8}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {15 \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{32 \sqrt {b}\, \sqrt {a +b}}+\frac {15 \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{32 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{3}}}{d}\)	\(299\)
risch	\(-\frac {8 a^{4} {\mathrm e}^{8 d x +8 c}+9 a^{3} b \,{\mathrm e}^{8 d x +8 c}+24 a^{2} b^{2} {\mathrm e}^{8 d x +8 c}+8 a \,b^{3} {\mathrm e}^{8 d x +8 c}+32 a^{4} {\mathrm e}^{6 d x +6 c}+82 a^{3} b \,{\mathrm e}^{6 d x +6 c}+54 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+80 a \,b^{3} {\mathrm e}^{6 d x +6 c}+16 b^{4} {\mathrm e}^{6 d x +6 c}+48 a^{4} {\mathrm e}^{4 d x +4 c}+128 a^{3} b \,{\mathrm e}^{4 d x +4 c}+106 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}-80 a \,b^{3} {\mathrm e}^{4 d x +4 c}-16 b^{4} {\mathrm e}^{4 d x +4 c}+32 a^{4} {\mathrm e}^{2 d x +2 c}+46 a^{3} b \,{\mathrm e}^{2 d x +2 c}-54 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-8 a \,b^{3} {\mathrm e}^{2 d x +2 c}+8 a^{4}-9 a^{3} b -2 a^{2} b^{2}}{4 a^{2} d \left (a +b \right )^{3} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{16 \left (a +b \right )^{4} d}-\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{16 \left (a +b \right )^{4} d}\)	\(454\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)-1/2/(a+b)^3/tanh(1/2*d*x+1/2*c)-2*b/(a+b)^3*(((-9/8*a-
9/8*b)*tanh(1/2*d*x+1/2*c)^7+(-27/8*a+1/8*b)*tanh(1/2*d*x+1/2*c)^5+(-27/8*a+1/8*b)*tanh(1/2*d*x+1/2*c)^3+(-9/8
*a-9/8*b)*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)^2-15/32/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))+15/32/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))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (116) = 232\).
time = 0.56, size = 533, normalized size = 4.23 \begin {gather*} -\frac {15 \, b \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 )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {8 \, a^{4} - 9 \, a^{3} b - 2 \, a^{2} b^{2} + 2 \, {\left (16 \, a^{4} + 23 \, a^{3} b - 27 \, a^{2} b^{2} - 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (24 \, a^{4} + 64 \, a^{3} b + 53 \, a^{2} b^{2} - 40 \, a b^{3} - 8 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (16 \, a^{4} + 41 \, a^{3} b + 27 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (8 \, a^{4} + 9 \, a^{3} b + 24 \, a^{2} b^{2} + 8 \, a b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} 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)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-15/16*b*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*sqrt((a + b)
*b)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) - 1/4*(8*a^4 - 9*a^3*b - 2*a^2*b^2 + 2*(16*a^4 + 23*
a^3*b - 27*a^2*b^2 - 4*a*b^3)*e^(-2*d*x - 2*c) + 2*(24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*e^(-4*d
*x - 4*c) + 2*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*e^(-6*d*x - 6*c) + (8*a^4 + 9*a^3*b + 24*a^2
*b^2 + 8*a*b^3)*e^(-8*d*x - 8*c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*
a^4*b^3 + 8*a^3*b^4)*e^(-2*d*x - 2*c) + 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e
^(-4*d*x - 4*c) - 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e^(-6*d*x - 6*c) - (3*a
^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*e^(-8*d*x - 8*c) - (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*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. 3499 vs. \(2 (116) = 232\).
time = 0.44, size = 7275, normalized size = 57.74 \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)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^8 + 32*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^
3)*cosh(d*x + c)*sinh(d*x + c)^7 + 4*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*sinh(d*x + c)^8 + 8*(16*a^4 + 41
*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*cosh(d*x + c)^6 + 8*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4
+ 14*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(14*(8*a^4 + 9*a^3*b + 24
*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^3 + 3*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*cosh(d*x + c))*sin
h(d*x + c)^5 + 8*(24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*cosh(d*x + c)^4 + 8*(35*(8*a^4 + 9*a^3*b
+ 24*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^4 + 24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4 + 15*(16*a^4 + 41*
a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*a^4 - 36*a^3*b - 8*a^2*b^2 + 32*(
7*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^5 + 5*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*
b^4)*cosh(d*x + c)^3 + (24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*
(16*a^4 + 23*a^3*b - 27*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2 + 8*(14*(8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*co
sh(d*x + c)^6 + 15*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 16*a^4 + 23*a^3*b - 2
7*a^2*b^2 - 4*a*b^3 + 6*(24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 -
15*(a^4*cosh(d*x + c)^10 + 10*a^4*cosh(d*x + c)*sinh(d*x + c)^9 + a^4*sinh(d*x + c)^10 + (3*a^4 + 8*a^3*b)*co
sh(d*x + c)^8 + (45*a^4*cosh(d*x + c)^2 + 3*a^4 + 8*a^3*b)*sinh(d*x + c)^8 + 8*(15*a^4*cosh(d*x + c)^3 + (3*a^
4 + 8*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 2*(105*a^4*cosh(
d*x + c)^4 + a^4 + 4*a^3*b + 8*a^2*b^2 + 14*(3*a^4 + 8*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*a^4*cos
h(d*x + c)^5 + 14*(3*a^4 + 8*a^3*b)*cosh(d*x + c)^3 + 3*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x +
c)^5 - 2*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 2*(105*a^4*cosh(d*x + c)^6 + 35*(3*a^4 + 8*a^3*b)*cosh(
d*x + c)^4 - a^4 - 4*a^3*b - 8*a^2*b^2 + 15*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4
+ 8*(15*a^4*cosh(d*x + c)^7 + 7*(3*a^4 + 8*a^3*b)*cosh(d*x + c)^5 + 5*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x +
c)^3 - (a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (3*a^4 + 8*a^3*b)*cosh(d*x + c)^2 + (45*a^
4*cosh(d*x + c)^8 + 28*(3*a^4 + 8*a^3*b)*cosh(d*x + c)^6 + 30*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 - 3*
a^4 - 8*a^3*b - 12*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*a^4*cosh(d*x + c)^9 + 4
*(3*a^4 + 8*a^3*b)*cosh(d*x + c)^7 + 6*(a^4 + 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 - 4*(a^4 + 4*a^3*b + 8*a^2*
b^2)*cosh(d*x + c)^3 - (3*a^4 + 8*a^3*b)*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*cos
h(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)*cos
h(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*(2*(8*a^4 + 9*a^3*b + 24*a^2*b^2 +
8*a*b^3)*cosh(d*x + c)^7 + 3*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*cosh(d*x + c)^5 + 2*(24*a^4 +
64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*cosh(d*x + c)^3 + (16*a^4 + 23*a^3*b - 27*a^2*b^2 - 4*a*b^3)*cosh(d
*x + c))*sinh(d*x + c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^10 + 10*(a^7 + 3*a^6*b + 3*a^5*
b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^10 + (3
*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^8 + (45*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^
4*b^3)*d*cosh(d*x + c)^2 + (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^8 + 2*(a^
7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^7 + 3*a^6*b + 3*a
^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)
)*sinh(d*x + c)^7 + 2*(105*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^4 + 14*(3*a^7 + 17*a^6*b + 33
*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 +
8*a^2*b^5)*d)*sinh(d*x + c)^6 - 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*d*cosh(d
*x + c)^4 + 4*(63*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 + 14*(3*a^7 + 17*a^6*b + 33*a^5*b^2
+ 27*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 + 3*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*
b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*...

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (116) = 232\).
time = 0.86, size = 347, normalized size = 2.75 \begin {gather*} \frac {\frac {15 \, b \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 {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 78 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 88 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 56 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 2 \, a^{2} b^{2}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{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 )}^{2}} - \frac {16}{{\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 )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(15*b*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)) - 2*(9*a^3*b*e^(6*d*x + 6*c) + 24*a^2*b^2*e^(6*d*x + 6*c) + 8*a*b^3*e^(6*d*x + 6*c) + 27*a^3*b*e^(4*
d*x + 4*c) + 78*a^2*b^2*e^(4*d*x + 4*c) + 88*a*b^3*e^(4*d*x + 4*c) + 16*b^4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*
x + 2*c) + 56*a^2*b^2*e^(2*d*x + 2*c) + 8*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b + 2*a^2*b^2)/((a^5 + 3*a^4*b + 3*a^3
*b^2 + a^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)^2) - 16/((a^3 + 3*a^2*b +
3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)))/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 )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cosh(c + d*x)^6/(sinh(c + d*x)^2*(b + a*cosh(c + d*x)^2)^3), x)

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