3.159 \(\int \text{csch}^6(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)
Optimal. Leaf size=88 \[ -\frac{a^2 \coth ^5(c+d x)}{5 d}+\frac{2 a^2 \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a b \coth (c+d x) \text{csch}(c+d x)}{d}+b^2 x \]
[Out]
b^2*x + (a*b*ArcTanh[Cosh[c + d*x]])/d - (a^2*Coth[c + d*x])/d + (2*a^2*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c +
d*x]^5)/(5*d) - (a*b*Coth[c + d*x]*Csch[c + d*x])/d
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Rubi [A] time = 0.102543, antiderivative size = 88, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3220, 3768, 3770, 3767} \[ -\frac{a^2 \coth ^5(c+d x)}{5 d}+\frac{2 a^2 \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a b \coth (c+d x) \text{csch}(c+d x)}{d}+b^2 x \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
b^2*x + (a*b*ArcTanh[Cosh[c + d*x]])/d - (a^2*Coth[c + d*x])/d + (2*a^2*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c +
d*x]^5)/(5*d) - (a*b*Coth[c + d*x]*Csch[c + d*x])/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 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]
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]
Rubi steps
\begin{align*} \int \text{csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\int \left (-b^2-2 a b \text{csch}^3(c+d x)-a^2 \text{csch}^6(c+d x)\right ) \, dx\\ &=b^2 x+a^2 \int \text{csch}^6(c+d x) \, dx+(2 a b) \int \text{csch}^3(c+d x) \, dx\\ &=b^2 x-\frac{a b \coth (c+d x) \text{csch}(c+d x)}{d}-(a b) \int \text{csch}(c+d x) \, dx-\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=b^2 x+\frac{a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a^2 \coth (c+d x)}{d}+\frac{2 a^2 \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth ^5(c+d x)}{5 d}-\frac{a b \coth (c+d x) \text{csch}(c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 0.941217, size = 197, normalized size = 2.24 \[ \frac{16 \left (-16 a^2 \tanh \left (\frac{1}{2} (c+d x)\right )-12 a^2 \sinh ^6\left (\frac{1}{2} (c+d x)\right ) \text{csch}^5(c+d x)-19 a^2 \sinh ^4\left (\frac{1}{2} (c+d x)\right ) \text{csch}^3(c+d x)-15 a b \text{sech}^2\left (\frac{1}{2} (c+d x)\right )-60 a b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+60 b^2 c+60 b^2 d x\right )-256 a^2 \coth \left (\frac{1}{2} (c+d x)\right )-3 a^2 \sinh (c+d x) \text{csch}^6\left (\frac{1}{2} (c+d x)\right )+19 a^2 \sinh (c+d x) \text{csch}^4\left (\frac{1}{2} (c+d x)\right )-240 a b \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{960 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(-256*a^2*Coth[(c + d*x)/2] - 240*a*b*Csch[(c + d*x)/2]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[c + d*x] - 3*a^2*C
sch[(c + d*x)/2]^6*Sinh[c + d*x] + 16*(60*b^2*c + 60*b^2*d*x - 60*a*b*Log[Tanh[(c + d*x)/2]] - 15*a*b*Sech[(c
+ d*x)/2]^2 - 19*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 12*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 16*a^2
*Tanh[(c + d*x)/2]))/(960*d)
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Maple [A] time = 0.108, size = 73, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right )+2\,ab \left ( -1/2\,{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^2,x)
[Out]
1/d*(a^2*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)+2*a*b*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(
exp(d*x+c)))+b^2*(d*x+c))
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Maxima [B] time = 1.1531, size = 409, normalized size = 4.65 \begin{align*} b^{2} x + a b{\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} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac{16}{15} \, a^{2}{\left (\frac{5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac{10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac{1}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
b^2*x + a*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(
-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 16/15*a^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x
- 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2
*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d
*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1
)))
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Fricas [B] time = 2.31189, size = 6268, normalized size = 71.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
1/15*(15*b^2*d*x*cosh(d*x + c)^10 + 15*b^2*d*x*sinh(d*x + c)^10 - 75*b^2*d*x*cosh(d*x + c)^8 - 30*a*b*cosh(d*x
+ c)^9 + 150*b^2*d*x*cosh(d*x + c)^6 + 30*(5*b^2*d*x*cosh(d*x + c) - a*b)*sinh(d*x + c)^9 + 60*a*b*cosh(d*x +
c)^7 + 15*(45*b^2*d*x*cosh(d*x + c)^2 - 5*b^2*d*x - 18*a*b*cosh(d*x + c))*sinh(d*x + c)^8 + 60*(30*b^2*d*x*co
sh(d*x + c)^3 - 10*b^2*d*x*cosh(d*x + c) - 18*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^7 + 30*(105*b^2*d*x*cos
h(d*x + c)^4 - 70*b^2*d*x*cosh(d*x + c)^2 - 84*a*b*cosh(d*x + c)^3 + 5*b^2*d*x + 14*a*b*cosh(d*x + c))*sinh(d*
x + c)^6 + 60*(63*b^2*d*x*cosh(d*x + c)^5 - 70*b^2*d*x*cosh(d*x + c)^3 - 63*a*b*cosh(d*x + c)^4 + 15*b^2*d*x*c
osh(d*x + c) + 21*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^5 - 60*a*b*cosh(d*x + c)^3 - 10*(15*b^2*d*x + 16*a^2)*cos
h(d*x + c)^4 + 10*(315*b^2*d*x*cosh(d*x + c)^6 - 525*b^2*d*x*cosh(d*x + c)^4 - 378*a*b*cosh(d*x + c)^5 + 225*b
^2*d*x*cosh(d*x + c)^2 + 210*a*b*cosh(d*x + c)^3 - 15*b^2*d*x - 16*a^2)*sinh(d*x + c)^4 - 15*b^2*d*x + 20*(90*
b^2*d*x*cosh(d*x + c)^7 - 210*b^2*d*x*cosh(d*x + c)^5 - 126*a*b*cosh(d*x + c)^6 + 150*b^2*d*x*cosh(d*x + c)^3
+ 105*a*b*cosh(d*x + c)^4 - 3*a*b - 2*(15*b^2*d*x + 16*a^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 30*a*b*cosh(d*x +
c) + 5*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^2 + 5*(135*b^2*d*x*cosh(d*x + c)^8 - 420*b^2*d*x*cosh(d*x + c)^6 -
216*a*b*cosh(d*x + c)^7 + 450*b^2*d*x*cosh(d*x + c)^4 + 252*a*b*cosh(d*x + c)^5 + 15*b^2*d*x - 36*a*b*cosh(d*
x + c) - 12*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^2 + 16*a^2)*sinh(d*x + c)^2 - 16*a^2 + 15*(a*b*cosh(d*x + c)^1
0 + 10*a*b*cosh(d*x + c)*sinh(d*x + c)^9 + a*b*sinh(d*x + c)^10 - 5*a*b*cosh(d*x + c)^8 + 5*(9*a*b*cosh(d*x +
c)^2 - a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x + c)^6 + 40*(3*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x
+ c)^7 + 10*(21*a*b*cosh(d*x + c)^4 - 14*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^6 - 10*a*b*cosh(d*x + c)^4
+ 4*(63*a*b*cosh(d*x + c)^5 - 70*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a*b*cosh
(d*x + c)^6 - 35*a*b*cosh(d*x + c)^4 + 15*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^4 + 5*a*b*cosh(d*x + c)^2 +
40*(3*a*b*cosh(d*x + c)^7 - 7*a*b*cosh(d*x + c)^5 + 5*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)^
3 + 5*(9*a*b*cosh(d*x + c)^8 - 28*a*b*cosh(d*x + c)^6 + 30*a*b*cosh(d*x + c)^4 - 12*a*b*cosh(d*x + c)^2 + a*b)
*sinh(d*x + c)^2 - a*b + 10*(a*b*cosh(d*x + c)^9 - 4*a*b*cosh(d*x + c)^7 + 6*a*b*cosh(d*x + c)^5 - 4*a*b*cosh(
d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 15*(a*b*cosh(d*x + c)^
10 + 10*a*b*cosh(d*x + c)*sinh(d*x + c)^9 + a*b*sinh(d*x + c)^10 - 5*a*b*cosh(d*x + c)^8 + 5*(9*a*b*cosh(d*x +
c)^2 - a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x + c)^6 + 40*(3*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*
x + c)^7 + 10*(21*a*b*cosh(d*x + c)^4 - 14*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^6 - 10*a*b*cosh(d*x + c)^4
+ 4*(63*a*b*cosh(d*x + c)^5 - 70*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a*b*cos
h(d*x + c)^6 - 35*a*b*cosh(d*x + c)^4 + 15*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^4 + 5*a*b*cosh(d*x + c)^2
+ 40*(3*a*b*cosh(d*x + c)^7 - 7*a*b*cosh(d*x + c)^5 + 5*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)
^3 + 5*(9*a*b*cosh(d*x + c)^8 - 28*a*b*cosh(d*x + c)^6 + 30*a*b*cosh(d*x + c)^4 - 12*a*b*cosh(d*x + c)^2 + a*b
)*sinh(d*x + c)^2 - a*b + 10*(a*b*cosh(d*x + c)^9 - 4*a*b*cosh(d*x + c)^7 + 6*a*b*cosh(d*x + c)^5 - 4*a*b*cosh
(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 10*(15*b^2*d*x*cosh(d
*x + c)^9 - 60*b^2*d*x*cosh(d*x + c)^7 - 27*a*b*cosh(d*x + c)^8 + 90*b^2*d*x*cosh(d*x + c)^5 + 42*a*b*cosh(d*x
+ c)^6 - 18*a*b*cosh(d*x + c)^2 - 4*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^3 + 3*a*b + (15*b^2*d*x + 16*a^2)*cos
h(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 - 5*d
*cosh(d*x + c)^8 + 5*(9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^8 + 40*(3*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*si
nh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 10*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 +
4*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 - 10*d*cosh(d*x + c)^4 +
10*(21*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 40*(3*d*cosh(d*x
+ c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*d*cosh(d*x + c)^2 +
5*(9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c
)^2 + 10*(d*cosh(d*x + c)^9 - 4*d*cosh(d*x + c)^7 + 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c
))*sinh(d*x + c) - d)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**6*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.24754, size = 197, normalized size = 2.24 \begin{align*} \frac{{\left (d x + c\right )} b^{2}}{d} + \frac{a b \log \left (e^{\left (d x + c\right )} + 1\right )}{d} - \frac{a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{2 \,{\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} - 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 8 \, a^{2}\right )}}{15 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
(d*x + c)*b^2/d + a*b*log(e^(d*x + c) + 1)/d - a*b*log(abs(e^(d*x + c) - 1))/d - 2/15*(15*a*b*e^(9*d*x + 9*c)
- 30*a*b*e^(7*d*x + 7*c) + 80*a^2*e^(4*d*x + 4*c) + 30*a*b*e^(3*d*x + 3*c) - 40*a^2*e^(2*d*x + 2*c) - 15*a*b*e
^(d*x + c) + 8*a^2)/(d*(e^(2*d*x + 2*c) - 1)^5)