3.1.58 \(\int \frac {\text {csch}^3(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [58]
Optimal. Leaf size=224 \[ \frac {b^{3/2} \left (35 a^2-56 a b+24 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^4 (a-b)^{5/2} d}+\frac {(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac {(2 a-3 b) b \cosh (c+d x)}{4 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(a-4 b) (4 a-3 b) b \cosh (c+d x)}{8 a^3 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2} \]
[Out]
1/8*b^(3/2)*(35*a^2-56*a*b+24*b^2)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/a^4/(a-b)^(5/2)/d+1/2*(a+6*b)*arcta
nh(cosh(d*x+c))/a^4/d-1/4*(2*a-3*b)*b*cosh(d*x+c)/a^2/(a-b)/d/(a-b+b*cosh(d*x+c)^2)^2-1/8*(a-4*b)*(4*a-3*b)*b*
cosh(d*x+c)/a^3/(a-b)^2/d/(a-b+b*cosh(d*x+c)^2)-1/2*coth(d*x+c)*csch(d*x+c)/a/d/(a-b+b*cosh(d*x+c)^2)^2
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Rubi [A]
time = 0.29, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3265, 425, 541,
536, 212, 211} \begin {gather*} \frac {(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac {b (a-4 b) (4 a-3 b) \cosh (c+d x)}{8 a^3 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {b (2 a-3 b) \cosh (c+d x)}{4 a^2 d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}+\frac {b^{3/2} \left (35 a^2-56 a b+24 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^4 d (a-b)^{5/2}}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b \cosh ^2(c+d x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
(b^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(5/2)*d) + ((a
+ 6*b)*ArcTanh[Cosh[c + d*x]])/(2*a^4*d) - ((2*a - 3*b)*b*Cosh[c + d*x])/(4*a^2*(a - b)*d*(a - b + b*Cosh[c +
d*x]^2)^2) - ((a - 4*b)*(4*a - 3*b)*b*Cosh[c + d*x])/(8*a^3*(a - b)^2*d*(a - b + b*Cosh[c + d*x]^2)) - (Coth[
c + d*x]*Csch[c + d*x])/(2*a*d*(a - b + b*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 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 3265
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {a+b+5 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{2 a d}\\ &=-\frac {(2 a-3 b) b \cosh (c+d x)}{4 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-2 \left (2 a^2+4 a b-3 b^2\right )-6 (2 a-3 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a-b) d}\\ &=-\frac {(2 a-3 b) b \cosh (c+d x)}{4 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(a-4 b) (4 a-3 b) b \cosh (c+d x)}{8 a^3 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {2 \left (4 a^3+12 a^2 b-25 a b^2+12 b^3\right )+2 (a-4 b) (4 a-3 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{16 a^3 (a-b)^2 d}\\ &=-\frac {(2 a-3 b) b \cosh (c+d x)}{4 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(a-4 b) (4 a-3 b) b \cosh (c+d x)}{8 a^3 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {(a+6 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^4 d}+\frac {\left (b^2 \left (35 a^2-56 a b+24 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^4 (a-b)^2 d}\\ &=\frac {b^{3/2} \left (35 a^2-56 a b+24 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^4 (a-b)^{5/2} d}+\frac {(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac {(2 a-3 b) b \cosh (c+d x)}{4 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(a-4 b) (4 a-3 b) b \cosh (c+d x)}{8 a^3 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.81, size = 419, normalized size = 1.87 \begin {gather*} \frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^5(c+d x) \left (\frac {8 a^2 b^2 \coth (c+d x)}{a-b}+\frac {2 a (11 a-8 b) b^2 (2 a-b+b \cosh (2 (c+d x))) \coth (c+d x)}{(a-b)^2}+\frac {b^{3/2} \left (35 a^2-56 a b+24 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right ) (2 a-b+b \cosh (2 (c+d x)))^2 \text {csch}(c+d x)}{(a-b)^{5/2}}+\frac {b^{3/2} \left (35 a^2-56 a b+24 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right ) (2 a-b+b \cosh (2 (c+d x)))^2 \text {csch}(c+d x)}{(a-b)^{5/2}}-a (2 a-b+b \cosh (2 (c+d x)))^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {csch}(c+d x)-4 (a+6 b) (2 a-b+b \cosh (2 (c+d x)))^2 \text {csch}(c+d x) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-a (2 a-b+b \cosh (2 (c+d x)))^2 \text {csch}(c+d x) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{64 a^4 d \left (b+a \text {csch}^2(c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^5*((8*a^2*b^2*Coth[c + d*x])/(a - b) + (2*a*(11*a - 8*b)*b^2*(2
*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x])/(a - b)^2 + (b^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b] -
I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x])/(a - b)^(5/2) + (b
^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cos
h[2*(c + d*x)])^2*Csch[c + d*x])/(a - b)^(5/2) - a*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Csch[
c + d*x] - 4*(a + 6*b)*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]] - a*(2*a - b + b
*Cosh[2*(c + d*x)])^2*Csch[c + d*x]*Sech[(c + d*x)/2]^2))/(64*a^4*d*(b + a*Csch[c + d*x]^2)^3)
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Maple [A]
time = 1.94, size = 350, normalized size = 1.56 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
[Out]
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a^3-1/8/a^3/tanh(1/2*d*x+1/2*c)^2+1/4/a^4*(-12*b-2*a)*ln(tanh(1/2*d*x+1/2*c))+4
*b^2/a^4*((-1/16*(13*a^2-40*a*b+24*b^2)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+1/16*(39*a^3-134*a^2*b+184*a*b
^2-80*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-1/16*a*(39*a^2-104*a*b+56*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1
/2*c)^2+1/16*a^2*(13*a-10*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+1/32*(35*a^2-56*a*b+24*b^2)/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
-1/4*((4*a^2*b^2*e^(11*c) - 19*a*b^3*e^(11*c) + 12*b^4*e^(11*c))*e^(11*d*x) + (32*a^3*b*e^(9*c) - 128*a^2*b^2*
e^(9*c) + 129*a*b^3*e^(9*c) - 36*b^4*e^(9*c))*e^(9*d*x) + 2*(32*a^4*e^(7*c) - 80*a^3*b*e^(7*c) + 94*a^2*b^2*e^
(7*c) - 55*a*b^3*e^(7*c) + 12*b^4*e^(7*c))*e^(7*d*x) + 2*(32*a^4*e^(5*c) - 80*a^3*b*e^(5*c) + 94*a^2*b^2*e^(5*
c) - 55*a*b^3*e^(5*c) + 12*b^4*e^(5*c))*e^(5*d*x) + (32*a^3*b*e^(3*c) - 128*a^2*b^2*e^(3*c) + 129*a*b^3*e^(3*c
) - 36*b^4*e^(3*c))*e^(3*d*x) + (4*a^2*b^2*e^c - 19*a*b^3*e^c + 12*b^4*e^c)*e^(d*x))/(a^5*b^2*d - 2*a^4*b^3*d
+ a^3*b^4*d + (a^5*b^2*d*e^(12*c) - 2*a^4*b^3*d*e^(12*c) + a^3*b^4*d*e^(12*c))*e^(12*d*x) + 2*(4*a^6*b*d*e^(10
*c) - 11*a^5*b^2*d*e^(10*c) + 10*a^4*b^3*d*e^(10*c) - 3*a^3*b^4*d*e^(10*c))*e^(10*d*x) + (16*a^7*d*e^(8*c) - 6
4*a^6*b*d*e^(8*c) + 95*a^5*b^2*d*e^(8*c) - 62*a^4*b^3*d*e^(8*c) + 15*a^3*b^4*d*e^(8*c))*e^(8*d*x) - 4*(8*a^7*d
*e^(6*c) - 28*a^6*b*d*e^(6*c) + 37*a^5*b^2*d*e^(6*c) - 22*a^4*b^3*d*e^(6*c) + 5*a^3*b^4*d*e^(6*c))*e^(6*d*x) +
(16*a^7*d*e^(4*c) - 64*a^6*b*d*e^(4*c) + 95*a^5*b^2*d*e^(4*c) - 62*a^4*b^3*d*e^(4*c) + 15*a^3*b^4*d*e^(4*c))*
e^(4*d*x) + 2*(4*a^6*b*d*e^(2*c) - 11*a^5*b^2*d*e^(2*c) + 10*a^4*b^3*d*e^(2*c) - 3*a^3*b^4*d*e^(2*c))*e^(2*d*x
)) + 1/2*(a + 6*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d) - 1/2*(a + 6*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4*d)
+ 8*integrate(1/32*((35*a^2*b^2*e^(3*c) - 56*a*b^3*e^(3*c) + 24*b^4*e^(3*c))*e^(3*d*x) - (35*a^2*b^2*e^c - 56
*a*b^3*e^c + 24*b^4*e^c)*e^(d*x))/(a^6*b - 2*a^5*b^2 + a^4*b^3 + (a^6*b*e^(4*c) - 2*a^5*b^2*e^(4*c) + a^4*b^3*
e^(4*c))*e^(4*d*x) + 2*(2*a^7*e^(2*c) - 5*a^6*b*e^(2*c) + 4*a^5*b^2*e^(2*c) - a^4*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. 12128 vs.
\(2 (206) = 412\).
time = 0.69, size = 22563, normalized size = 100.73 \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*sinh(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
[-1/16*(4*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^11 + 44*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*cosh(d
*x + c)*sinh(d*x + c)^10 + 4*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*sinh(d*x + c)^11 + 4*(32*a^4*b - 128*a^3*b^2
+ 129*a^2*b^3 - 36*a*b^4)*cosh(d*x + c)^9 + 4*(32*a^4*b - 128*a^3*b^2 + 129*a^2*b^3 - 36*a*b^4 + 55*(4*a^3*b^2
- 19*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 12*(55*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*cosh(d
*x + c)^3 + 3*(32*a^4*b - 128*a^3*b^2 + 129*a^2*b^3 - 36*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^8 + 8*(32*a^5 - 8
0*a^4*b + 94*a^3*b^2 - 55*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^7 + 8*(32*a^5 - 80*a^4*b + 94*a^3*b^2 - 55*a^2*b^3
+ 12*a*b^4 + 165*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^4 + 18*(32*a^4*b - 128*a^3*b^2 + 129*a^2*b
^3 - 36*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 56*(33*(4*a^3*b^2 - 19*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^5 +
6*(32*a^4*b - 128*a^3*b^2 + 129*a^2*b^3 - 36*a*b^4)*cosh(d*x + c)^3 + (32*a^5 - 80*a^4*b + 94*a^3*b^2 - 55*a^
2*b^3 + 12 ...
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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)**3/(a+b*sinh(d*x+c)**2)**3,x)
[Out]
Timed out
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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*sinh(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 {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\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)^3*(a + b*sinh(c + d*x)^2)^3),x)
[Out]
int(1/(sinh(c + d*x)^3*(a + b*sinh(c + d*x)^2)^3), x)
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