3.1.47 \(\int \frac {\text {csch}^3(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [47]

Optimal. Leaf size=213 \[ -\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^4 d}+\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2} \]

[Out]

1/2*(a-5*b)*arctanh(cosh(d*x+c))/(a+b)^4/d+1/4*(2*a-b)*b*cosh(d*x+c)/a/(a+b)^2/d/(b+a*cosh(d*x+c)^2)^2-1/8*(4*
a^2-9*a*b-b^2)*cosh(d*x+c)/a/(a+b)^3/d/(b+a*cosh(d*x+c)^2)-1/2*cosh(d*x+c)*coth(d*x+c)^2/(a+b)/d/(b+a*cosh(d*x
+c)^2)^2-1/8*(15*a^2-10*a*b-b^2)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^(3/2)/(a+b)^4/d

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Rubi [A]
time = 0.25, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4218, 481, 592, 541, 536, 212, 211} \begin {gather*} -\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} d (a+b)^4}+\frac {b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(a^(3/2)*(a + b)^4*d) + ((a - 5
*b)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^4*d) + ((2*a - b)*b*Cosh[c + d*x])/(4*a*(a + b)^2*d*(b + a*Cosh[c + d*x
]^2)^2) - ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(8*a*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - (Cosh[c + d*x]*Cot
h[c + d*x]^2)/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2)^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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, 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 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 4218

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 b+(-a+2 b) x^2\right )}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{2 (a+b) d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {2 (2 a-b) b-2 \left (2 a^2-8 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^2 d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-2 (11 a-b) b^2+2 b \left (4 a^2-9 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{16 a b (a+b)^3 d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {(a-5 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^4 d}-\frac {\left (b \left (15 a^2-10 a b-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^4 d}\\ &=-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^4 d}+\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.79, size = 524, normalized size = 2.46 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^5(c+d x) \left (-\frac {8 b^2 (a+b)^2}{a}+\frac {2 b (a+b) (9 a+b) (a+2 b+a \cosh (2 (c+d x)))}{a}+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x)}{a^{3/2}}+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x)}{a^{3/2}}-(a+b) (a+2 b+a \cosh (2 (c+d x)))^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)+4 (a-5 b) (a+2 b+a \cosh (2 (c+d x)))^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-4 (a-5 b) (a+2 b+a \cosh (2 (c+d x)))^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-(a+b) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)\right )}{64 (a+b)^4 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]^3/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((-8*b^2*(a + b)^2)/a + (2*b*(a + b)*(9*a + b)*(a + 2*b + a*C
osh[2*(c + d*x)]))/a + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh
[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/
Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[(
(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]
*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2)
- (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 5*b)*(a + 2*b + a*Cosh
[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh
[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(
64*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [A]
time = 2.85, size = 350, normalized size = 1.64

method	result	size
derivativedivides	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}+24 a^{2} b +24 a \,b^{2}+8 b^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{3}-5 a^{2} b -13 a \,b^{2}+b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (27 a^{3}-21 a^{2} b +29 a \,b^{2}-3 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (27 a^{3}+a^{2} b -23 a \,b^{2}+3 b^{3}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {9 a^{3}+17 a^{2} b +7 a \,b^{2}-b^{3}}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (15 a^{2}-10 a b -b^{2}\right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{16 a \sqrt {a b}}\right )}{\left (a +b \right )^{4}}-\frac {1}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +10 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{4}}}{d}\)	\(350\)
default	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}+24 a^{2} b +24 a \,b^{2}+8 b^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{3}-5 a^{2} b -13 a \,b^{2}+b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (27 a^{3}-21 a^{2} b +29 a \,b^{2}-3 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (27 a^{3}+a^{2} b -23 a \,b^{2}+3 b^{3}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {9 a^{3}+17 a^{2} b +7 a \,b^{2}-b^{3}}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (15 a^{2}-10 a b -b^{2}\right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{16 a \sqrt {a b}}\right )}{\left (a +b \right )^{4}}-\frac {1}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +10 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{4}}}{d}\)	\(350\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (4 a^{3} {\mathrm e}^{10 d x +10 c}-9 a^{2} b \,{\mathrm e}^{10 d x +10 c}-a \,b^{2} {\mathrm e}^{10 d x +10 c}+20 a^{3} {\mathrm e}^{8 d x +8 c}+23 a^{2} b \,{\mathrm e}^{8 d x +8 c}-29 a \,b^{2} {\mathrm e}^{8 d x +8 c}+4 b^{3} {\mathrm e}^{8 d x +8 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+114 a^{2} b \,{\mathrm e}^{6 d x +6 c}+94 a \,b^{2} {\mathrm e}^{6 d x +6 c}-4 b^{3} {\mathrm e}^{6 d x +6 c}+40 a^{3} {\mathrm e}^{4 d x +4 c}+114 a^{2} b \,{\mathrm e}^{4 d x +4 c}+94 a \,b^{2} {\mathrm e}^{4 d x +4 c}-4 b^{3} {\mathrm e}^{4 d x +4 c}+20 a^{3} {\mathrm e}^{2 d x +2 c}+23 a^{2} b \,{\mathrm e}^{2 d x +2 c}-29 a \,b^{2} {\mathrm e}^{2 d x +2 c}+4 b^{3} {\mathrm e}^{2 d x +2 c}+4 a^{3}-9 a^{2} b -a \,b^{2}\right )}{4 a d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \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}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) a}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {5 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) a}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {5 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{16 \left (a +b \right )^{4} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{8 a \left (a +b \right )^{4} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{16 \left (a +b \right )^{4} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{8 a \left (a +b \right )^{4} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} d}\)	\(833\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)-2*b/(a+b)^4*((-1/8*(9*a^3-5*a^2*b-13*a*b^2+b^3)/a*tan
h(1/2*d*x+1/2*c)^6-1/8*(27*a^3-21*a^2*b+29*a*b^2-3*b^3)/a*tanh(1/2*d*x+1/2*c)^4-1/8*(27*a^3+a^2*b-23*a*b^2+3*b
^3)/a*tanh(1/2*d*x+1/2*c)^2-1/8*(9*a^3+17*a^2*b+7*a*b^2-b^3)/a)/(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+1/16*(15*a^2-10*a*b-b^2)/a/(a*b)^(1/2)*arctan(1/
4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2)))-1/8/(a+b)^3/tanh(1/2*d*x+1/2*c)^2+1/4/(a+b)^4*(-2*a+10
*b)*ln(tanh(1/2*d*x+1/2*c)))

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

[Out]

1/2*(a - 5*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/2*(a - 5
*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/4*((4*a^3*e^(11*c)
- 9*a^2*b*e^(11*c) - a*b^2*e^(11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 23*a^2*b*e^(9*c) - 29*a*b^2*e^(9*c) + 4*b
^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 57*a^2*b*e^(7*c) + 47*a*b^2*e^(7*c) - 2*b^3*e^(7*c))*e^(7*d*x) + 2
*(20*a^3*e^(5*c) + 57*a^2*b*e^(5*c) + 47*a*b^2*e^(5*c) - 2*b^3*e^(5*c))*e^(5*d*x) + (20*a^3*e^(3*c) + 23*a^2*b
*e^(3*c) - 29*a*b^2*e^(3*c) + 4*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c - 9*a^2*b*e^c - a*b^2*e^c)*e^(d*x))/(a^6*d
+ 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d*e^(12*c) + a^3*b^3
*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) + 7*a^5*b*d*e^(10*c) + 15*a^4*b^2*d*e^(10*c) + 13*a^3*b^3*d*e^(10*
c) + 4*a^2*b^4*d*e^(10*c))*e^(10*d*x) - (a^6*d*e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^4*b^2*d*e^(8*c) - 47*a^3*b^3
*d*e^(8*c) - 48*a^2*b^4*d*e^(8*c) - 16*a*b^5*d*e^(8*c))*e^(8*d*x) - 4*(a^6*d*e^(6*c) + 7*a^5*b*d*e^(6*c) + 23*
a^4*b^2*d*e^(6*c) + 37*a^3*b^3*d*e^(6*c) + 28*a^2*b^4*d*e^(6*c) + 8*a*b^5*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c
) + 3*a^5*b*d*e^(4*c) - 13*a^4*b^2*d*e^(4*c) - 47*a^3*b^3*d*e^(4*c) - 48*a^2*b^4*d*e^(4*c) - 16*a*b^5*d*e^(4*c
))*e^(4*d*x) + 2*(a^6*d*e^(2*c) + 7*a^5*b*d*e^(2*c) + 15*a^4*b^2*d*e^(2*c) + 13*a^3*b^3*d*e^(2*c) + 4*a^2*b^4*
d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/32*((15*a^2*b*e^(3*c) - 10*a*b^2*e^(3*c) - b^3*e^(3*c))*e^(3*d*x) - (15*
a^2*b*e^c - 10*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 + (a^6*e^(4*c) +
4*a^5*b*e^(4*c) + 6*a^4*b^2*e^(4*c) + 4*a^3*b^3*e^(4*c) + a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 6*a^5
*b*e^(2*c) + 14*a^4*b^2*e^(2*c) + 16*a^3*b^3*e^(2*c) + 9*a^2*b^4*e^(2*c) + 2*a*b^5*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. 10990 vs. \(2 (195) = 390\).
time = 0.58, size = 20341, normalized size = 95.50 \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)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^11 + 44*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*
cosh(d*x + c)*sinh(d*x + c)^10 + 4*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*sinh(d*x + c)^11 + 4*(20*a^4 + 43*a^
3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^9 + 4*(20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4 + 55
*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 12*(55*(4*a^4 - 5*a^3*b - 10*a^2*b^
2 - a*b^3)*cosh(d*x + c)^3 + 3*(20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c)
^8 + 8*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c)^7 + 8*(165*(4*a^4 - 5*a^3*b - 10*a^2
*b^2 - a*b^3)*cosh(d*x + c)^4 + 20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4 + 18*(20*a^4 + 43*a^3*b - 6
*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 56*(33*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*
cosh(d*x + c)^5 + 6*(20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^3 + (20*a^4 + 77*a^3*b +
104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c)^6 + 8*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3
- 2*b^4)*cosh(d*x + c)^5 + 8*(231*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^6 + 63*(20*a^4 + 43*a^
3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^4 + 20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4 + 21*
(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 8*(165*(4*a^4 - 5*a^3*
b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^7 + 63*(20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^
5 + 35*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c)^3 + 5*(20*a^4 + 77*a^3*b + 104*a^2*b
^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*c
osh(d*x + c)^3 + 4*(165*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 84*(20*a^4 + 43*a^3*b - 6*a^2
*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^6 + 70*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x +
c)^4 + 20*a^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4 + 20*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b
^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(55*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cosh(d*x + c)^9 + 36*(20*a
^4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c)^7 + 42*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3
- 2*b^4)*cosh(d*x + c)^5 + 20*(20*a^4 + 77*a^3*b + 104*a^2*b^2 + 45*a*b^3 - 2*b^4)*cosh(d*x + c)^3 + 3*(20*a^
4 + 43*a^3*b - 6*a^2*b^2 - 25*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^4 - 10*a^3*b - a^2*b^2)*c
osh(d*x + c)^12 + 12*(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (15*a^4 - 10*a^3*b - a^2*b
^2)*sinh(d*x + c)^12 + 2*(15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^10 + 2*(15*a^4 + 50*a^3*b -
41*a^2*b^2 - 4*a*b^3 + 33*(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(15*a^4 - 1
0*a^3*b - a^2*b^2)*cosh(d*x + c)^3 + (15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^9
- (15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*cosh(d*x + c)^8 + (495*(15*a^4 - 10*a^3*b - a^2*b^2)
*cosh(d*x + c)^4 - 15*a^4 + 10*a^3*b + 241*a^2*b^2 - 160*a*b^3 - 16*b^4 + 90*(15*a^4 + 50*a^3*b - 41*a^2*b^2 -
4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^5 + 30*(15*a^4
+ 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 - (15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*co
sh(d*x + c))*sinh(d*x + c)^7 - 4*(15*a^4 + 50*a^3*b + 79*a^2*b^2 - 84*a*b^3 - 8*b^4)*cosh(d*x + c)^6 + 4*(231*
(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^6 + 105*(15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^4
- 15*a^4 - 50*a^3*b - 79*a^2*b^2 + 84*a*b^3 + 8*b^4 - 7*(15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(99*(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^7 + 63*(15*a^4 + 50*a^3
*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^5 - 7*(15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*cosh(d*x
+ c)^3 - 3*(15*a^4 + 50*a^3*b + 79*a^2*b^2 - 84*a*b^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - (15*a^4 - 10*
a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*cosh(d*x + c)^4 + (495*(15*a^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^8
+ 420*(15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^6 - 70*(15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*
a*b^3 + 16*b^4)*cosh(d*x + c)^4 - 15*a^4 + 10*a^3*b + 241*a^2*b^2 - 160*a*b^3 - 16*b^4 - 60*(15*a^4 + 50*a^3*b
+ 79*a^2*b^2 - 84*a*b^3 - 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^4 - 10*a^3*b - a^2*b^2 + 4*(55*(15*a
^4 - 10*a^3*b - a^2*b^2)*cosh(d*x + c)^9 + 60*(15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^7 - 14*
(15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*cosh(d*x + c)^5 - 20*(15*a^4 + 50*a^3*b + 79*a^2*b^2 -
84*a*b^3 - 8*b^4)*cosh(d*x + c)^3 - (15*a^4 - 10*a^3*b - 241*a^2*b^2 + 160*a*b^3 + 16*b^4)*cosh(d*x + c))*sinh
(d*x + c)^3 + 2*(15*a^4 + 50*a^3*b - 41*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2 + 2*(33*(15*a^4 - 10*a^3*b - a^2*b^
2)*cosh(d*x + c)^10 + 45*(15*a^4 + 50*a^3*b - 4...

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

[Out]

Integral(csch(c + d*x)**3/(a + b*sech(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)^3/(a+b*sech(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 = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\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)^3*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

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

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