3.156 \(\int \text {csch}^3(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)
Optimal. Leaf size=77 \[ \frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}+2 a b x+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {b^2 \cosh (c+d x)}{d} \]
[Out]
2*a*b*x+1/2*a^2*arctanh(cosh(d*x+c))/d-b^2*cosh(d*x+c)/d+1/3*b^2*cosh(d*x+c)^3/d-1/2*a^2*coth(d*x+c)*csch(d*x+
c)/d
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Rubi [A] time = 0.11, antiderivative size = 77, normalized size of antiderivative = 1.00,
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, 2633} \[ \frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}+2 a b x+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {b^2 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
2*a*b*x + (a^2*ArcTanh[Cosh[c + d*x]])/(2*d) - (b^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d) - (a^2*Coth
[c + d*x]*Csch[c + d*x])/(2*d)
Rule 2633
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (2 i a b+i a^2 \text {csch}^3(c+d x)+i b^2 \sinh ^3(c+d x)\right ) \, dx\right )\\ &=2 a b x+a^2 \int \text {csch}^3(c+d x) \, dx+b^2 \int \sinh ^3(c+d x) \, dx\\ &=2 a b x-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {1}{2} a^2 \int \text {csch}(c+d x) \, dx-\frac {b^2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=2 a b x+\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 105, normalized size = 1.36 \[ -\frac {a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+2 a b x-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
2*a*b*x - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^
2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d)
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fricas [B] time = 0.68, size = 1616, normalized size = 20.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
1/24*(b^2*cosh(d*x + c)^10 + 10*b^2*cosh(d*x + c)*sinh(d*x + c)^9 + b^2*sinh(d*x + c)^10 + 48*a*b*d*x*cosh(d*x
+ c)^7 - 11*b^2*cosh(d*x + c)^8 - 96*a*b*d*x*cosh(d*x + c)^5 + (45*b^2*cosh(d*x + c)^2 - 11*b^2)*sinh(d*x + c
)^8 + 8*(15*b^2*cosh(d*x + c)^3 + 6*a*b*d*x - 11*b^2*cosh(d*x + c))*sinh(d*x + c)^7 + 48*a*b*d*x*cosh(d*x + c)
^3 - 2*(12*a^2 - 5*b^2)*cosh(d*x + c)^6 + 2*(105*b^2*cosh(d*x + c)^4 + 168*a*b*d*x*cosh(d*x + c) - 154*b^2*cos
h(d*x + c)^2 - 12*a^2 + 5*b^2)*sinh(d*x + c)^6 + 4*(63*b^2*cosh(d*x + c)^5 + 252*a*b*d*x*cosh(d*x + c)^2 - 154
*b^2*cosh(d*x + c)^3 - 24*a*b*d*x - 3*(12*a^2 - 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(12*a^2 - 5*b^2)*cos
h(d*x + c)^4 + 2*(105*b^2*cosh(d*x + c)^6 + 840*a*b*d*x*cosh(d*x + c)^3 - 385*b^2*cosh(d*x + c)^4 - 240*a*b*d*
x*cosh(d*x + c) - 15*(12*a^2 - 5*b^2)*cosh(d*x + c)^2 - 12*a^2 + 5*b^2)*sinh(d*x + c)^4 - 11*b^2*cosh(d*x + c)
^2 + 8*(15*b^2*cosh(d*x + c)^7 + 210*a*b*d*x*cosh(d*x + c)^4 - 77*b^2*cosh(d*x + c)^5 - 120*a*b*d*x*cosh(d*x +
c)^2 + 6*a*b*d*x - 5*(12*a^2 - 5*b^2)*cosh(d*x + c)^3 - (12*a^2 - 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (45
*b^2*cosh(d*x + c)^8 + 1008*a*b*d*x*cosh(d*x + c)^5 - 308*b^2*cosh(d*x + c)^6 - 960*a*b*d*x*cosh(d*x + c)^3 +
144*a*b*d*x*cosh(d*x + c) - 30*(12*a^2 - 5*b^2)*cosh(d*x + c)^4 - 12*(12*a^2 - 5*b^2)*cosh(d*x + c)^2 - 11*b^2
)*sinh(d*x + c)^2 + b^2 + 12*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7
- 2*a^2*cosh(d*x + c)^5 + (21*a^2*cosh(d*x + c)^2 - 2*a^2)*sinh(d*x + c)^5 + a^2*cosh(d*x + c)^3 + 5*(7*a^2*co
sh(d*x + c)^3 - 2*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x + c)^2 + a^2)
*sinh(d*x + c)^3 + (21*a^2*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (
7*a^2*cosh(d*x + c)^6 - 10*a^2*cosh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sin
h(d*x + c) + 1) - 12*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7 - 2*a^2*
cosh(d*x + c)^5 + (21*a^2*cosh(d*x + c)^2 - 2*a^2)*sinh(d*x + c)^5 + a^2*cosh(d*x + c)^3 + 5*(7*a^2*cosh(d*x +
c)^3 - 2*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*
x + c)^3 + (21*a^2*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (7*a^2*co
sh(d*x + c)^6 - 10*a^2*cosh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x +
c) - 1) + 2*(5*b^2*cosh(d*x + c)^9 + 168*a*b*d*x*cosh(d*x + c)^6 - 44*b^2*cosh(d*x + c)^7 - 240*a*b*d*x*cosh(d
*x + c)^4 + 72*a*b*d*x*cosh(d*x + c)^2 - 6*(12*a^2 - 5*b^2)*cosh(d*x + c)^5 - 4*(12*a^2 - 5*b^2)*cosh(d*x + c)
^3 - 11*b^2*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x
+ c)^7 - 2*d*cosh(d*x + c)^5 + (21*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^5 + 5*(7*d*cosh(d*x + c)^3 - 2*d*cos
h(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (35*d*cosh(d*x + c)^4 - 20*d*cosh(d*x + c)^2 + d)*sinh(d*x +
c)^3 + (21*d*cosh(d*x + c)^5 - 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*d*cosh(d*x + c)
^6 - 10*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c))
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giac [B] time = 0.20, size = 162, normalized size = 2.10 \[ \frac {48 \, {\left (d x + c\right )} a b + b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (d x + c\right )} + 12 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 12 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {{\left (11 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2} + 3 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} + {\left (24 \, a^{2} - 19 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{2} {\left (e^{\left (d x + c\right )} - 1\right )}^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
1/24*(48*(d*x + c)*a*b + b^2*e^(3*d*x + 3*c) - 9*b^2*e^(d*x + c) + 12*a^2*log(e^(d*x + c) + 1) - 12*a^2*log(ab
s(e^(d*x + c) - 1)) - (11*b^2*e^(2*d*x + 2*c) - b^2 + 3*(8*a^2 + 3*b^2)*e^(6*d*x + 6*c) + (24*a^2 - 19*b^2)*e^
(4*d*x + 4*c))*e^(-3*d*x - 3*c)/((e^(d*x + c) + 1)^2*(e^(d*x + c) - 1)^2))/d
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maple [A] time = 0.13, size = 63, normalized size = 0.82 \[ \frac {a^{2} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (d x +c \right )+b^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3*(a+b*sinh(d*x+c)^3)^2,x)
[Out]
1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(d*x+c)+b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*
x+c))
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maxima [B] time = 0.33, size = 152, normalized size = 1.97 \[ 2 \, a b x + \frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, a^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
2*a*b*x + 1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 1/2*a^2*(lo
g(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)))
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mupad [B] time = 0.71, size = 175, normalized size = 2.27 \[ \frac {\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}-\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}+2\,a\,b\,x-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^3)^2/sinh(c + d*x)^3,x)
[Out]
(atan((a^2*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^4)^(1/2)))*(a^4)^(1/2))/(-d^2)^(1/2) - (3*b^2*exp(c + d*x))/(8*
d) + 2*a*b*x - (3*b^2*exp(- c - d*x))/(8*d) + (b^2*exp(- 3*c - 3*d*x))/(24*d) + (b^2*exp(3*c + 3*d*x))/(24*d)
- (a^2*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^2*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*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)**3*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
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