3.157 \(\int \text{csch}^4(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)
Optimal. Leaf size=76 \[ -\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^2 x}{2} \]
[Out]
-(b^2*x)/2 - (2*a*b*ArcTanh[Cosh[c + d*x]])/d + (a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) + (b^2*Cos
h[c + d*x]*Sinh[c + d*x])/(2*d)
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Rubi [A] time = 0.0910175, antiderivative size = 76, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{3220, 3770, 3767, 2635, 8} \[ -\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^2 x}{2} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
-(b^2*x)/2 - (2*a*b*ArcTanh[Cosh[c + d*x]])/d + (a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) + (b^2*Cos
h[c + d*x]*Sinh[c + d*x])/(2*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 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]
Rule 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \text{csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=\int \left (2 a b \text{csch}(c+d x)+a^2 \text{csch}^4(c+d x)+b^2 \sinh ^2(c+d x)\right ) \, dx\\ &=a^2 \int \text{csch}^4(c+d x) \, dx+(2 a b) \int \text{csch}(c+d x) \, dx+b^2 \int \sinh ^2(c+d x) \, dx\\ &=-\frac{2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{1}{2} b^2 \int 1 \, dx+\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac{b^2 x}{2}-\frac{2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{a^2 \coth (c+d x)}{d}-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.398911, size = 81, normalized size = 1.07 \[ \frac{3 b \left (b \sinh (2 (c+d x))-2 \left (-4 a \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )+4 a \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )+b c+b d x\right )\right )-4 a^2 \coth (c+d x) \left (\text{csch}^2(c+d x)-2\right )}{12 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(-4*a^2*Coth[c + d*x]*(-2 + Csch[c + d*x]^2) + 3*b*(-2*(b*c + b*d*x + 4*a*Log[Cosh[(c + d*x)/2]] - 4*a*Log[Sin
h[(c + d*x)/2]]) + b*Sinh[2*(c + d*x)]))/(12*d)
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Maple [A] time = 0.069, size = 65, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )-4\,ab{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +{b}^{2} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^4*(a+b*sinh(d*x+c)^3)^2,x)
[Out]
1/d*(a^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-4*a*b*arctanh(exp(d*x+c))+b^2*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*
x-1/2*c))
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Maxima [B] time = 1.25967, size = 230, normalized size = 3.03 \begin{align*} -\frac{1}{8} \, b^{2}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - 2 \, 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}\right )} + \frac{4}{3} \, a^{2}{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^4*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
-1/8*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 2*a*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) -
1)/d) + 4/3*a^2*(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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Fricas [B] time = 2.08082, size = 4551, normalized size = 59.88 \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)^4*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
1/24*(3*b^2*cosh(d*x + c)^10 + 30*b^2*cosh(d*x + c)*sinh(d*x + c)^9 + 3*b^2*sinh(d*x + c)^10 - 3*(4*b^2*d*x +
3*b^2)*cosh(d*x + c)^8 - 3*(4*b^2*d*x - 45*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^8 + 24*(15*b^2*cosh(d*x
+ c)^3 - (4*b^2*d*x + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 6*(6*b^2*d*x + b^2)*cosh(d*x + c)^6 + 6*(105*b^2
*cosh(d*x + c)^4 + 6*b^2*d*x - 14*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^6 + 12*(63*b^2*cosh
(d*x + c)^5 - 14*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)^3 + 3*(6*b^2*d*x + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*
(6*b^2*d*x + 16*a^2 - b^2)*cosh(d*x + c)^4 + 6*(105*b^2*cosh(d*x + c)^6 - 35*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)
^4 - 6*b^2*d*x + 15*(6*b^2*d*x + b^2)*cosh(d*x + c)^2 - 16*a^2 + b^2)*sinh(d*x + c)^4 + 24*(15*b^2*cosh(d*x +
c)^7 - 7*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)^5 + 5*(6*b^2*d*x + b^2)*cosh(d*x + c)^3 - (6*b^2*d*x + 16*a^2 - b^2
)*cosh(d*x + c))*sinh(d*x + c)^3 + (12*b^2*d*x + 32*a^2 - 9*b^2)*cosh(d*x + c)^2 + (135*b^2*cosh(d*x + c)^8 -
84*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)^6 + 90*(6*b^2*d*x + b^2)*cosh(d*x + c)^4 + 12*b^2*d*x - 36*(6*b^2*d*x + 1
6*a^2 - b^2)*cosh(d*x + c)^2 + 32*a^2 - 9*b^2)*sinh(d*x + c)^2 + 3*b^2 - 48*(a*b*cosh(d*x + c)^8 + 8*a*b*cosh(
d*x + c)*sinh(d*x + c)^7 + a*b*sinh(d*x + c)^8 - 3*a*b*cosh(d*x + c)^6 + (28*a*b*cosh(d*x + c)^2 - 3*a*b)*sinh
(d*x + c)^6 + 3*a*b*cosh(d*x + c)^4 + 2*(28*a*b*cosh(d*x + c)^3 - 9*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + (70*a
*b*cosh(d*x + c)^4 - 45*a*b*cosh(d*x + c)^2 + 3*a*b)*sinh(d*x + c)^4 - a*b*cosh(d*x + c)^2 + 4*(14*a*b*cosh(d*
x + c)^5 - 15*a*b*cosh(d*x + c)^3 + 3*a*b*cosh(d*x + c))*sinh(d*x + c)^3 + (28*a*b*cosh(d*x + c)^6 - 45*a*b*co
sh(d*x + c)^4 + 18*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 + 2*(4*a*b*cosh(d*x + c)^7 - 9*a*b*cosh(d*x + c)
^5 + 6*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) + 48*(a*
b*cosh(d*x + c)^8 + 8*a*b*cosh(d*x + c)*sinh(d*x + c)^7 + a*b*sinh(d*x + c)^8 - 3*a*b*cosh(d*x + c)^6 + (28*a*
b*cosh(d*x + c)^2 - 3*a*b)*sinh(d*x + c)^6 + 3*a*b*cosh(d*x + c)^4 + 2*(28*a*b*cosh(d*x + c)^3 - 9*a*b*cosh(d*
x + c))*sinh(d*x + c)^5 + (70*a*b*cosh(d*x + c)^4 - 45*a*b*cosh(d*x + c)^2 + 3*a*b)*sinh(d*x + c)^4 - a*b*cosh
(d*x + c)^2 + 4*(14*a*b*cosh(d*x + c)^5 - 15*a*b*cosh(d*x + c)^3 + 3*a*b*cosh(d*x + c))*sinh(d*x + c)^3 + (28*
a*b*cosh(d*x + c)^6 - 45*a*b*cosh(d*x + c)^4 + 18*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 + 2*(4*a*b*cosh(d
*x + c)^7 - 9*a*b*cosh(d*x + c)^5 + 6*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) + 2*(15*b^2*cosh(d*x + c)^9 - 12*(4*b^2*d*x + 3*b^2)*cosh(d*x + c)^7 + 18*(6*b^2*d*x +
b^2)*cosh(d*x + c)^5 - 12*(6*b^2*d*x + 16*a^2 - b^2)*cosh(d*x + c)^3 + (12*b^2*d*x + 32*a^2 - 9*b^2)*cosh(d*x
+ c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 3*d*cosh(d*x
+ c)^6 + (28*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^6 + 2*(28*d*cosh(d*x + c)^3 - 9*d*cosh(d*x + c))*sinh(d*x
+ c)^5 + 3*d*cosh(d*x + c)^4 + (70*d*cosh(d*x + c)^4 - 45*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 4*(14*d*
cosh(d*x + c)^5 - 15*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - d*cosh(d*x + c)^2 + (28*d*cosh(d
*x + c)^6 - 45*d*cosh(d*x + c)^4 + 18*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 2*(4*d*cosh(d*x + c)^7 - 9*d*co
sh(d*x + c)^5 + 6*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c))
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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)**4*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
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Giac [B] time = 1.25275, size = 219, normalized size = 2.88 \begin{align*} -\frac{{\left (d x + c\right )} b^{2}}{2 \, d} + \frac{b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac{2 \, a b \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{2 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{{\left (3 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{2} + 3 \,{\left (32 \, a^{2} - 3 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} -{\left (32 \, a^{2} - 9 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{24 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{3}{\left (e^{\left (d x + c\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^4*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
-1/2*(d*x + c)*b^2/d + 1/8*b^2*e^(2*d*x + 2*c)/d - 2*a*b*log(e^(d*x + c) + 1)/d + 2*a*b*log(abs(e^(d*x + c) -
1))/d - 1/24*(3*b^2*e^(6*d*x + 6*c) - 3*b^2 + 3*(32*a^2 - 3*b^2)*e^(4*d*x + 4*c) - (32*a^2 - 9*b^2)*e^(2*d*x +
2*c))*e^(-2*d*x - 2*c)/(d*(e^(d*x + c) + 1)^3*(e^(d*x + c) - 1)^3)