3.1.59 \(\int \frac {\text {csch}^4(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [59]

Optimal. Leaf size=259 \[ \frac {b^2 \left (48 a^2-80 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} (a-b)^{5/2} d}+\frac {\left (8 a^3-4 a^2 b-45 a b^2+35 b^3\right ) \coth (c+d x)}{8 a^4 (a-b)^2 d}-\frac {\left (8 a^2-52 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a-b)^2 d}-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(10 a-7 b) b \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

1/8*b^2*(48*a^2-80*a*b+35*b^2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(9/2)/(a-b)^(5/2)/d+1/8*(8*a^3-4*a^2
*b-45*a*b^2+35*b^3)*coth(d*x+c)/a^4/(a-b)^2/d-1/24*(8*a^2-52*a*b+35*b^2)*coth(d*x+c)^3/a^3/(a-b)^2/d-1/4*b*csc
h(d*x+c)^3*sech(d*x+c)^3/a/(a-b)/d/(a-(a-b)*tanh(d*x+c)^2)^2-1/8*(10*a-7*b)*b*csch(d*x+c)^3*sech(d*x+c)/a^2/(a
-b)^2/d/(a-(a-b)*tanh(d*x+c)^2)

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Rubi [A]
time = 0.26, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3266, 479, 591, 584, 214} \begin {gather*} -\frac {b (10 a-7 b) \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {b^2 \left (48 a^2-80 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d (a-b)^{5/2}}-\frac {\left (8 a^2-52 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 d (a-b)^2}+\frac {\left (8 a^3-4 a^2 b-45 a b^2+35 b^3\right ) \coth (c+d x)}{8 a^4 d (a-b)^2}-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(b^2*(48*a^2 - 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^(5/2)*d) + ((
8*a^3 - 4*a^2*b - 45*a*b^2 + 35*b^3)*Coth[c + d*x])/(8*a^4*(a - b)^2*d) - ((8*a^2 - 52*a*b + 35*b^2)*Coth[c +
d*x]^3)/(24*a^3*(a - b)^2*d) - (b*Csch[c + d*x]^3*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2
)^2) - ((10*a - 7*b)*b*Csch[c + d*x]^3*Sech[c + d*x])/(8*a^2*(a - b)^2*d*(a - (a - 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 479

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

Rule 584

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

Rule 591

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 3266

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^4 \left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (4 a-7 b+(-4 a+b) x^2\right )}{x^4 \left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(10 a-7 b) b \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (8 a^2-52 a b+35 b^2+\left (-8 a^2+12 a b-7 b^2\right ) x^2\right )}{x^4 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(10 a-7 b) b \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \left (\frac {8 a^2-52 a b+35 b^2}{a x^4}+\frac {-8 a^3+4 a^2 b+45 a b^2-35 b^3}{a^2 x^2}+\frac {b^2 \left (48 a^2-80 a b+35 b^2\right )}{a^2 \left (a-(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=\frac {\left (8 a^3-4 a^2 b-45 a b^2+35 b^3\right ) \coth (c+d x)}{8 a^4 (a-b)^2 d}-\frac {\left (8 a^2-52 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a-b)^2 d}-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(10 a-7 b) b \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\left (b^2 \left (48 a^2-80 a b+35 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 (a-b)^2 d}\\ &=\frac {b^2 \left (48 a^2-80 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} (a-b)^{5/2} d}+\frac {\left (8 a^3-4 a^2 b-45 a b^2+35 b^3\right ) \coth (c+d x)}{8 a^4 (a-b)^2 d}-\frac {\left (8 a^2-52 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a-b)^2 d}-\frac {b \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(10 a-7 b) b \text {csch}^3(c+d x) \text {sech}(c+d x)}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.87, size = 167, normalized size = 0.64 \begin {gather*} \frac {\frac {3 b^2 \left (48 a^2-80 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a-b)^{5/2}}+\sqrt {a} \left (-8 \coth (c+d x) \left (-2 a-9 b+a \text {csch}^2(c+d x)\right )+\frac {3 b^3 \left (-32 a^2+40 a b-11 b^2+b (-14 a+11 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}\right )}{24 a^{9/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((3*b^2*(48*a^2 - 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(5/2) + Sqrt[a]*(-8*C
oth[c + d*x]*(-2*a - 9*b + a*Csch[c + d*x]^2) + (3*b^3*(-32*a^2 + 40*a*b - 11*b^2 + b*(-14*a + 11*b)*Cosh[2*(c
+ d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2)))/(24*a^(9/2)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(496\) vs. \(2(241)=482\).
time = 1.87, size = 497, normalized size = 1.92 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8/a^4*(1/3*a*tanh(1/2*d*x+1/2*c)^3-3*a*tanh(1/2*d*x+1/2*c)-12*b*tanh(1/2*d*x+1/2*c))-1/24/a^3/tanh(1/2
*d*x+1/2*c)^3-1/8/a^4*(-12*b-3*a)/tanh(1/2*d*x+1/2*c)-4*b^2/a^4*((1/16*a*b*(16*a-13*b)/(a^2-2*a*b+b^2)*tanh(1/
2*d*x+1/2*c)^7-1/16*(16*a^2-69*a*b+44*b^2)*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-1/16*(16*a^2-69*a*b+44*b^2)
*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+1/16*a*b*(16*a-13*b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c))/(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*(48*a^2-80*a*b+35*b^2)/(a^2-2*a*b+
b^2)*a*(1/2*((-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x
+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)
+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8519 vs. \(2 (243) = 486\).
time = 0.54, size = 17294, normalized size = 66.77 \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*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/48*(12*(48*a^4*b^3 - 128*a^3*b^4 + 115*a^2*b^5 - 35*a*b^6)*cosh(d*x + c)^12 + 144*(48*a^4*b^3 - 128*a^3*b^4
+ 115*a^2*b^5 - 35*a*b^6)*cosh(d*x + c)*sinh(d*x + c)^11 + 12*(48*a^4*b^3 - 128*a^3*b^4 + 115*a^2*b^5 - 35*a*
b^6)*sinh(d*x + c)^12 + 72*(48*a^5*b^2 - 176*a^4*b^3 + 243*a^3*b^4 - 150*a^2*b^5 + 35*a*b^6)*cosh(d*x + c)^10
+ 72*(48*a^5*b^2 - 176*a^4*b^3 + 243*a^3*b^4 - 150*a^2*b^5 + 35*a*b^6 + 11*(48*a^4*b^3 - 128*a^3*b^4 + 115*a^2
*b^5 - 35*a*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 240*(11*(48*a^4*b^3 - 128*a^3*b^4 + 115*a^2*b^5 - 35*a*b^
6)*cosh(d*x + c)^3 + 3*(48*a^5*b^2 - 176*a^4*b^3 + 243*a^3*b^4 - 150*a^2*b^5 + 35*a*b^6)*cosh(d*x + c))*sinh(d
*x + c)^9 + 4*(768*a^6*b - 5408*a^5*b^2 + 12960*a^4*b^3 - 14370*a^3*b^4 + 7625*a^2*b^5 - 1575*a*b^6)*cosh(d*x
+ c)^8 + 4*(768*a^6*b - 5408*a^5*b^2 + 12960*a^4*b^3 - 14370*a^3*b^4 + 7625*a^2*b^5 - 1575*a*b^6 + 1485*(48*a^
4*b^3 - 128*a^3*b^4 + 115*a^2*b^5 - 35*a*b^6)*cosh(d*x + c)^4 + 810*(48*a^5*b^2 - 176*a^4*b^3 + 243*a^3*b^4 -
150*a^2*b^ ...

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

[Out]

Timed out

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Giac [A]
time = 0.82, size = 378, normalized size = 1.46 \begin {gather*} \frac {\frac {3 \, {\left (48 \, a^{2} b^{2} - 80 \, a b^{3} + 35 \, b^{4}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {6 \, {\left (24 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 32 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 11 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 112 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 200 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 130 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 33 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 88 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 112 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, a b^{4} - 11 \, b^{5}\right )}}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} + \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 18 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 9 \, b\right )}}{a^{4} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/24*(3*(48*a^2*b^2 - 80*a*b^3 + 35*b^4)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^6 - 2*
a^5*b + a^4*b^2)*sqrt(-a^2 + a*b)) + 6*(24*a^2*b^3*e^(6*d*x + 6*c) - 32*a*b^4*e^(6*d*x + 6*c) + 11*b^5*e^(6*d*
x + 6*c) + 112*a^3*b^2*e^(4*d*x + 4*c) - 200*a^2*b^3*e^(4*d*x + 4*c) + 130*a*b^4*e^(4*d*x + 4*c) - 33*b^5*e^(4
*d*x + 4*c) + 88*a^2*b^3*e^(2*d*x + 2*c) - 112*a*b^4*e^(2*d*x + 2*c) + 33*b^5*e^(2*d*x + 2*c) + 14*a*b^4 - 11*
b^5)/((a^6 - 2*a^5*b + a^4*b^2)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2) + 16*(9
*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2*c) - 18*b*e^(2*d*x + 2*c) + 2*a + 9*b)/(a^4*(e^(2*d*x + 2*c) - 1)^3))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\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)^4*(a + b*sinh(c + d*x)^2)^3),x)

[Out]

int(1/(sinh(c + d*x)^4*(a + b*sinh(c + d*x)^2)^3), x)

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