3.158 \(\int \text{csch}^5(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)
Optimal. Leaf size=90 \[ -\frac{3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{2 a b \coth (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d} \]
[Out]
(-3*a^2*ArcTanh[Cosh[c + d*x]])/(8*d) + (b^2*Cosh[c + d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Coth[c + d*x]
*Csch[c + d*x])/(8*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)
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Rubi [A] time = 0.132429, antiderivative size = 90, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used =
{3220, 3767, 8, 3768, 3770, 2638} \[ -\frac{3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{2 a b \coth (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(-3*a^2*ArcTanh[Cosh[c + d*x]])/(8*d) + (b^2*Cosh[c + d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Coth[c + d*x]
*Csch[c + d*x])/(8*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(4*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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \text{csch}^5(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-2 i a b \text{csch}^2(c+d x)-i a^2 \text{csch}^5(c+d x)-i b^2 \sinh (c+d x)\right ) \, dx\\ &=a^2 \int \text{csch}^5(c+d x) \, dx+(2 a b) \int \text{csch}^2(c+d x) \, dx+b^2 \int \sinh (c+d x) \, dx\\ &=\frac{b^2 \cosh (c+d x)}{d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}-\frac{1}{4} \left (3 a^2\right ) \int \text{csch}^3(c+d x) \, dx-\frac{(2 i a b) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac{b^2 \cosh (c+d x)}{d}-\frac{2 a b \coth (c+d x)}{d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{1}{8} \left (3 a^2\right ) \int \text{csch}(c+d x) \, dx\\ &=-\frac{3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac{b^2 \cosh (c+d x)}{d}-\frac{2 a b \coth (c+d x)}{d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0380619, size = 149, normalized size = 1.66 \[ -\frac{a^2 \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a^2 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{3 a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{2 a b \coth (c+d x)}{d}+\frac{b^2 \sinh (c) \sinh (d x)}{d}+\frac{b^2 \cosh (c) \cosh (d x)}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(b^2*Cosh[c]*Cosh[d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Csch[(c + d*x)/2]^2)/(32*d) - (a^2*Csch[(c + d*x)
/2]^4)/(64*d) + (3*a^2*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^2*Sech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x
)/2]^4)/(64*d) + (b^2*Sinh[c]*Sinh[d*x])/d
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Maple [A] time = 0.109, size = 66, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (dx+c\right )}{8}} \right ){\rm coth} \left (dx+c\right )-{\frac{3\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{4}} \right ) -2\,ab{\rm coth} \left (dx+c\right )+{b}^{2}\cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x)
[Out]
1/d*(a^2*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-2*a*b*coth(d*x+c)+b^2*cosh
(d*x+c))
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Maxima [B] time = 1.12204, size = 254, normalized size = 2.82 \begin{align*} \frac{1}{2} \, b^{2}{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} - \frac{1}{8} \, a^{2}{\left (\frac{3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + \frac{4 \, a b}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
1/2*b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) - 1/8*a^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*
(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e
^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + 4*a*b/(d*(e^(-2*d*x - 2*c) - 1))
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Fricas [B] time = 2.26968, size = 5495, normalized size = 61.06 \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)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
1/8*(4*b^2*cosh(d*x + c)^10 + 40*b^2*cosh(d*x + c)*sinh(d*x + c)^9 + 4*b^2*sinh(d*x + c)^10 - 32*a*b*cosh(d*x
+ c)^7 + 6*(a^2 - 2*b^2)*cosh(d*x + c)^8 + 6*(30*b^2*cosh(d*x + c)^2 + a^2 - 2*b^2)*sinh(d*x + c)^8 + 16*(30*b
^2*cosh(d*x + c)^3 - 2*a*b + 3*(a^2 - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 96*a*b*cosh(d*x + c)^5 - 2*(11*a
^2 - 4*b^2)*cosh(d*x + c)^6 + 2*(420*b^2*cosh(d*x + c)^4 - 112*a*b*cosh(d*x + c) + 84*(a^2 - 2*b^2)*cosh(d*x +
c)^2 - 11*a^2 + 4*b^2)*sinh(d*x + c)^6 + 12*(84*b^2*cosh(d*x + c)^5 - 56*a*b*cosh(d*x + c)^2 + 28*(a^2 - 2*b^
2)*cosh(d*x + c)^3 + 8*a*b - (11*a^2 - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 96*a*b*cosh(d*x + c)^3 - 2*(11*
a^2 - 4*b^2)*cosh(d*x + c)^4 + 2*(420*b^2*cosh(d*x + c)^6 - 560*a*b*cosh(d*x + c)^3 + 210*(a^2 - 2*b^2)*cosh(d
*x + c)^4 + 240*a*b*cosh(d*x + c) - 15*(11*a^2 - 4*b^2)*cosh(d*x + c)^2 - 11*a^2 + 4*b^2)*sinh(d*x + c)^4 + 8*
(60*b^2*cosh(d*x + c)^7 - 140*a*b*cosh(d*x + c)^4 + 42*(a^2 - 2*b^2)*cosh(d*x + c)^5 + 120*a*b*cosh(d*x + c)^2
- 5*(11*a^2 - 4*b^2)*cosh(d*x + c)^3 - 12*a*b - (11*a^2 - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 32*a*b*cosh
(d*x + c) + 6*(a^2 - 2*b^2)*cosh(d*x + c)^2 + 6*(30*b^2*cosh(d*x + c)^8 - 112*a*b*cosh(d*x + c)^5 + 28*(a^2 -
2*b^2)*cosh(d*x + c)^6 + 160*a*b*cosh(d*x + c)^3 - 5*(11*a^2 - 4*b^2)*cosh(d*x + c)^4 - 48*a*b*cosh(d*x + c) -
2*(11*a^2 - 4*b^2)*cosh(d*x + c)^2 + a^2 - 2*b^2)*sinh(d*x + c)^2 + 4*b^2 - 3*(a^2*cosh(d*x + c)^9 + 9*a^2*co
sh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x + c)^9 - 4*a^2*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 - a^2)*si
nh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 28*(3*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(21
*a^2*cosh(d*x + c)^4 - 14*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^5 - 4*a^2*cosh(d*x + c)^3 + 2*(63*a^2*cosh(
d*x + c)^5 - 70*a^2*cosh(d*x + c)^3 + 15*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*a^2*cosh(d*x + c)^6 - 35*a
^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^3 + a^2*cosh(d*x + c) + 12*(3*a^2*cosh(d*x +
c)^7 - 7*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x +
c)^8 - 28*a^2*cosh(d*x + c)^6 + 30*a^2*cosh(d*x + c)^4 - 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*log(cosh
(d*x + c) + sinh(d*x + c) + 1) + 3*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x +
c)^9 - 4*a^2*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 28*(
3*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(21*a^2*cosh(d*x + c)^4 - 14*a^2*cosh(d*x + c)^
2 + a^2)*sinh(d*x + c)^5 - 4*a^2*cosh(d*x + c)^3 + 2*(63*a^2*cosh(d*x + c)^5 - 70*a^2*cosh(d*x + c)^3 + 15*a^2
*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*a^2*cosh(d*x + c)^6 - 35*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2
- a^2)*sinh(d*x + c)^3 + a^2*cosh(d*x + c) + 12*(3*a^2*cosh(d*x + c)^7 - 7*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*
x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 - 28*a^2*cosh(d*x + c)^6 + 30*a^2*cosh(
d*x + c)^4 - 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(10*b^2*c
osh(d*x + c)^9 - 56*a*b*cosh(d*x + c)^6 + 12*(a^2 - 2*b^2)*cosh(d*x + c)^7 + 120*a*b*cosh(d*x + c)^4 - 3*(11*a
^2 - 4*b^2)*cosh(d*x + c)^5 - 72*a*b*cosh(d*x + c)^2 - 2*(11*a^2 - 4*b^2)*cosh(d*x + c)^3 + 8*a*b + 3*(a^2 - 2
*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + d*sinh(d*x + c)^9
- 4*d*cosh(d*x + c)^7 + 4*(9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^7 + 28*(3*d*cosh(d*x + c)^3 - d*cosh(d*x +
c))*sinh(d*x + c)^6 + 6*d*cosh(d*x + c)^5 + 6*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^
5 + 2*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^4 - 4*d*cosh(d*x + c)^3
+ 4*(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)^3 + 12*(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)^2 + d*cosh(d*x + c) + (
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))
________________________________________________________________________________________
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)**5*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
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Giac [B] time = 1.26709, size = 246, normalized size = 2.73 \begin{align*} \frac{b^{2} e^{\left (d x + c\right )}}{2 \, d} + \frac{b^{2} e^{\left (-d x - c\right )}}{2 \, d} - \frac{3 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{8 \, d} + \frac{3 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{8 \, d} + \frac{3 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 16 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 11 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 48 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 11 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} e^{\left (d x + c\right )} + 16 \, a b}{4 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
1/2*b^2*e^(d*x + c)/d + 1/2*b^2*e^(-d*x - c)/d - 3/8*a^2*log(e^(d*x + c) + 1)/d + 3/8*a^2*log(abs(e^(d*x + c)
- 1))/d + 1/4*(3*a^2*e^(7*d*x + 7*c) - 16*a*b*e^(6*d*x + 6*c) - 11*a^2*e^(5*d*x + 5*c) + 48*a*b*e^(4*d*x + 4*c
) - 11*a^2*e^(3*d*x + 3*c) - 48*a*b*e^(2*d*x + 2*c) + 3*a^2*e^(d*x + c) + 16*a*b)/(d*(e^(2*d*x + 2*c) - 1)^4)