3.160 \(\int \text {csch}^7(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)

Optimal. Leaf size=133 \[ \frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}-\frac {2 a b \coth ^3(c+d x)}{3 d}+\frac {2 a b \coth (c+d x)}{d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d} \]

[Out]

5/16*a^2*arctanh(cosh(d*x+c))/d-b^2*arctanh(cosh(d*x+c))/d+2*a*b*coth(d*x+c)/d-2/3*a*b*coth(d*x+c)^3/d-5/16*a^
2*coth(d*x+c)*csch(d*x+c)/d+5/24*a^2*coth(d*x+c)*csch(d*x+c)^3/d-1/6*a^2*coth(d*x+c)*csch(d*x+c)^5/d

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Rubi [A]  time = 0.17, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3220, 3770, 3767, 3768} \[ \frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}-\frac {2 a b \coth ^3(c+d x)}{3 d}+\frac {2 a b \coth (c+d x)}{d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2,x]

[Out]

(5*a^2*ArcTanh[Cosh[c + d*x]])/(16*d) - (b^2*ArcTanh[Cosh[c + d*x]])/d + (2*a*b*Coth[c + d*x])/d - (2*a*b*Coth
[c + d*x]^3)/(3*d) - (5*a^2*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d)
- (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (i b^2 \text {csch}(c+d x)+2 i a b \text {csch}^4(c+d x)+i a^2 \text {csch}^7(c+d x)\right ) \, dx\right )\\ &=a^2 \int \text {csch}^7(c+d x) \, dx+(2 a b) \int \text {csch}^4(c+d x) \, dx+b^2 \int \text {csch}(c+d x) \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {1}{6} \left (5 a^2\right ) \int \text {csch}^5(c+d x) \, dx+\frac {(2 i a b) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^2\right ) \int \text {csch}^3(c+d x) \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^2\right ) \int \text {csch}(c+d x) \, dx\\ &=\frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 235, normalized size = 1.77 \[ -\frac {a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {4 a b \coth (c+d x)}{3 d}-\frac {2 a b \coth (c+d x) \text {csch}^2(c+d x)}{3 d}+\frac {b^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2,x]

[Out]

(4*a*b*Coth[c + d*x])/(3*d) - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) + (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (a^2*Csc
h[(c + d*x)/2]^6)/(384*d) - (2*a*b*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b^2*Log[Cosh[c/2 + (d*x)/2]])/d + (
b^2*Log[Sinh[c/2 + (d*x)/2]])/d - (5*a^2*Log[Tanh[(c + d*x)/2]])/(16*d) - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) -
(a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d)

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fricas [B]  time = 0.52, size = 3607, normalized size = 27.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/48*(30*a^2*cosh(d*x + c)^11 + 330*a^2*cosh(d*x + c)*sinh(d*x + c)^10 + 30*a^2*sinh(d*x + c)^11 - 170*a^2*co
sh(d*x + c)^9 + 384*a*b*cosh(d*x + c)^8 + 10*(165*a^2*cosh(d*x + c)^2 - 17*a^2)*sinh(d*x + c)^9 + 396*a^2*cosh
(d*x + c)^7 + 6*(825*a^2*cosh(d*x + c)^3 - 255*a^2*cosh(d*x + c) + 64*a*b)*sinh(d*x + c)^8 - 1280*a*b*cosh(d*x
+ c)^6 + 12*(825*a^2*cosh(d*x + c)^4 - 510*a^2*cosh(d*x + c)^2 + 256*a*b*cosh(d*x + c) + 33*a^2)*sinh(d*x + c
)^7 + 396*a^2*cosh(d*x + c)^5 + 4*(3465*a^2*cosh(d*x + c)^5 - 3570*a^2*cosh(d*x + c)^3 + 2688*a*b*cosh(d*x + c
)^2 + 693*a^2*cosh(d*x + c) - 320*a*b)*sinh(d*x + c)^6 + 1536*a*b*cosh(d*x + c)^4 + 12*(1155*a^2*cosh(d*x + c)
^6 - 1785*a^2*cosh(d*x + c)^4 + 1792*a*b*cosh(d*x + c)^3 + 693*a^2*cosh(d*x + c)^2 - 640*a*b*cosh(d*x + c) + 3
3*a^2)*sinh(d*x + c)^5 - 170*a^2*cosh(d*x + c)^3 + 12*(825*a^2*cosh(d*x + c)^7 - 1785*a^2*cosh(d*x + c)^5 + 22
40*a*b*cosh(d*x + c)^4 + 1155*a^2*cosh(d*x + c)^3 - 1600*a*b*cosh(d*x + c)^2 + 165*a^2*cosh(d*x + c) + 128*a*b
)*sinh(d*x + c)^4 - 768*a*b*cosh(d*x + c)^2 + 2*(2475*a^2*cosh(d*x + c)^8 - 7140*a^2*cosh(d*x + c)^6 + 10752*a
*b*cosh(d*x + c)^5 + 6930*a^2*cosh(d*x + c)^4 - 12800*a*b*cosh(d*x + c)^3 + 1980*a^2*cosh(d*x + c)^2 + 3072*a*
b*cosh(d*x + c) - 85*a^2)*sinh(d*x + c)^3 + 30*a^2*cosh(d*x + c) + 6*(275*a^2*cosh(d*x + c)^9 - 1020*a^2*cosh(
d*x + c)^7 + 1792*a*b*cosh(d*x + c)^6 + 1386*a^2*cosh(d*x + c)^5 - 3200*a*b*cosh(d*x + c)^4 + 660*a^2*cosh(d*x
+ c)^3 + 1536*a*b*cosh(d*x + c)^2 - 85*a^2*cosh(d*x + c) - 128*a*b)*sinh(d*x + c)^2 + 128*a*b - 3*((5*a^2 - 1
6*b^2)*cosh(d*x + c)^12 + 12*(5*a^2 - 16*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^2 - 16*b^2)*sinh(d*x + c)^
12 - 6*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2)*sinh(d*x +
c)^10 + 20*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a
^2 - 16*b^2)*cosh(d*x + c)^8 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 - 18*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 +
5*a^2 - 16*b^2)*sinh(d*x + c)^8 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 30*(5*a^2 - 16*b^2)*cosh(d*x + c)
^3 + 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 4*(231*(5*a^2 -
16*b^2)*cosh(d*x + c)^6 - 315*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 105*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 25*a^
2 + 80*b^2)*sinh(d*x + c)^6 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 63*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 +
35*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^2 - 16*b^2)*
cosh(d*x + c)^4 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 - 84*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 70*(5*a^2 -
16*b^2)*cosh(d*x + c)^4 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 + 5*a^2 - 16*b^2)*sinh(d*x + c)^4 + 20*(11*(5*a^
2 - 16*b^2)*cosh(d*x + c)^9 - 36*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 + 42*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 20*(
5*a^2 - 16*b^2)*cosh(d*x + c)^3 + 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*(5*a^2 - 16*b^2)*cosh(
d*x + c)^2 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 - 45*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 + 70*(5*a^2 - 16*b^
2)*cosh(d*x + c)^6 - 50*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 15*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^
2)*sinh(d*x + c)^2 + 5*a^2 - 16*b^2 + 12*((5*a^2 - 16*b^2)*cosh(d*x + c)^11 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c)
^9 + 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 5*(5*a^2 - 16*b^2)*cosh(d*x +
c)^3 - (5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*((5*a^2 - 16
*b^2)*cosh(d*x + c)^12 + 12*(5*a^2 - 16*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^2 - 16*b^2)*sinh(d*x + c)^1
2 - 6*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2)*sinh(d*x +
c)^10 + 20*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a^
2 - 16*b^2)*cosh(d*x + c)^8 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 - 18*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 +
5*a^2 - 16*b^2)*sinh(d*x + c)^8 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 30*(5*a^2 - 16*b^2)*cosh(d*x + c)^
3 + 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 4*(231*(5*a^2 -
16*b^2)*cosh(d*x + c)^6 - 315*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 105*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 25*a^2
+ 80*b^2)*sinh(d*x + c)^6 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 63*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 3
5*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^2 - 16*b^2)*c
osh(d*x + c)^4 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 - 84*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 70*(5*a^2 - 1
6*b^2)*cosh(d*x + c)^4 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 + 5*a^2 - 16*b^2)*sinh(d*x + c)^4 + 20*(11*(5*a^2
- 16*b^2)*cosh(d*x + c)^9 - 36*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 + 42*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 20*(5
*a^2 - 16*b^2)*cosh(d*x + c)^3 + 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*(5*a^2 - 16*b^2)*cosh(d
*x + c)^2 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 - 45*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 + 70*(5*a^2 - 16*b^2
)*cosh(d*x + c)^6 - 50*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 15*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2
)*sinh(d*x + c)^2 + 5*a^2 - 16*b^2 + 12*((5*a^2 - 16*b^2)*cosh(d*x + c)^11 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c)^
9 + 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 5*(5*a^2 - 16*b^2)*cosh(d*x +
c)^3 - (5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(55*a^2*cosh(
d*x + c)^10 - 255*a^2*cosh(d*x + c)^8 + 512*a*b*cosh(d*x + c)^7 + 462*a^2*cosh(d*x + c)^6 - 1280*a*b*cosh(d*x
+ c)^5 + 330*a^2*cosh(d*x + c)^4 + 1024*a*b*cosh(d*x + c)^3 - 85*a^2*cosh(d*x + c)^2 - 256*a*b*cosh(d*x + c) +
5*a^2)*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 - 6*d*co
sh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))
*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 - 18*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8
+ 24*(33*d*cosh(d*x + c)^5 - 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 - 20*d*cosh(d*x + c)^6
+ 4*(231*d*cosh(d*x + c)^6 - 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^6 + 24*(33*d*
cosh(d*x + c)^7 - 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^5 + 15*d*cosh
(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 - 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 - 20*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 - 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 - 20*d*cosh(d*x +
c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 - 45*d*cosh(d*x + c
)^8 + 70*d*cosh(d*x + c)^6 - 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 12*(d*cosh(d*x
+ c)^11 - 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 - 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*
x + c))*sinh(d*x + c) + d)

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giac [A]  time = 0.25, size = 204, normalized size = 1.53 \[ \frac {3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{2} e^{\left (11 \, d x + 11 \, c\right )} - 85 \, a^{2} e^{\left (9 \, d x + 9 \, c\right )} + 192 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 198 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 198 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 768 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 85 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 384 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} e^{\left (d x + c\right )} + 64 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")

[Out]

1/48*(3*(5*a^2 - 16*b^2)*log(e^(d*x + c) + 1) - 3*(5*a^2 - 16*b^2)*log(abs(e^(d*x + c) - 1)) - 2*(15*a^2*e^(11
*d*x + 11*c) - 85*a^2*e^(9*d*x + 9*c) + 192*a*b*e^(8*d*x + 8*c) + 198*a^2*e^(7*d*x + 7*c) - 640*a*b*e^(6*d*x +
6*c) + 198*a^2*e^(5*d*x + 5*c) + 768*a*b*e^(4*d*x + 4*c) - 85*a^2*e^(3*d*x + 3*c) - 384*a*b*e^(2*d*x + 2*c) +
15*a^2*e^(d*x + c) + 64*a*b)/(e^(2*d*x + 2*c) - 1)^6)/d

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maple [A]  time = 0.18, size = 90, normalized size = 0.68 \[ \frac {a^{2} \left (\left (-\frac {\mathrm {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \mathrm {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\mathrm {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \arctanh \left ({\mathrm e}^{d x +c}\right )}{8}\right )+2 a b \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x)

[Out]

1/d*(a^2*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth(d*x+c)+5/8*arctanh(exp(d*x+c)))+2*a*b*
(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-2*b^2*arctanh(exp(d*x+c)))

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maxima [B]  time = 0.33, size = 316, normalized size = 2.38 \[ \frac {1}{48} \, a^{2} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} - b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {8}{3} \, a b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

1/48*a^2*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d + 2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3*c) +
198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x -
2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 1
2*c) - 1))) - b^2*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d) + 8/3*a*b*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-
2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e
^(-6*d*x - 6*c) - 1)))

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mupad [B]  time = 0.16, size = 434, normalized size = 3.26 \[ \frac {\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{12\,d}-\frac {8\,a\,b}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {16\,a\,b}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^2\,\sqrt {-d^2}-16\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}\right )\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}{8\,\sqrt {-d^2}}-\frac {18\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {80\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {32\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^3)^2/sinh(c + d*x)^7,x)

[Out]

((5*a^2*exp(c + d*x))/(12*d) - (8*a*b)/d)/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - ((a^2*exp(c + d*x))/(3
*d) + (16*a*b)/(3*d))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) + (atan((exp(d*x)*exp(c
)*(5*a^2*(-d^2)^(1/2) - 16*b^2*(-d^2)^(1/2)))/(d*(25*a^4 + 256*b^4 - 160*a^2*b^2)^(1/2)))*(25*a^4 + 256*b^4 -
160*a^2*b^2)^(1/2))/(8*(-d^2)^(1/2)) - (18*a^2*exp(c + d*x))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*e
xp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (80*a^2*exp(c + d*x))/(3*d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x
) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1)) - (32*a^2*exp(c + d*x))/(3*d*(15*exp(4
*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c
+ 12*d*x) + 1)) - (5*a^2*exp(c + d*x))/(8*d*(exp(2*c + 2*d*x) - 1))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**7*(a+b*sinh(d*x+c)**3)**2,x)

[Out]

Timed out

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