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]
-1/2*b^2*x-2*a*b*arctanh(cosh(d*x+c))/d+a^2*coth(d*x+c)/d-1/3*a^2*coth(d*x+c)^3/d+1/2*b^2*cosh(d*x+c)*sinh(d*x
+c)/d
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Rubi [A] time = 0.09, antiderivative size = 76, normalized size of antiderivative = 1.00,
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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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 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 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}^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*}
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Mathematica [A] time = 0.41, 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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fricas [B] time = 0.57, size = 1748, normalized size = 23.00 \[ \text {result too large to display} \]
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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giac [B] time = 0.20, size = 151, normalized size = 1.99 \[ -\frac {12 \, {\left (d x + c\right )} b^{2} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) - 48 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {{\left (3 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{2} + 3 \, {\left (32 \, a^{2} - 3 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (32 \, a^{2} - 9 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{3} {\left (e^{\left (d x + c\right )} - 1\right )}^{3}}}{24 \, d} \]
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]
-1/24*(12*(d*x + c)*b^2 - 3*b^2*e^(2*d*x + 2*c) + 48*a*b*log(e^(d*x + c) + 1) - 48*a*b*log(abs(e^(d*x + c) - 1
)) + (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)/((e^(d*x + c) + 1)^3*(e^(d*x + c) - 1)^3))/d
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maple [A] time = 0.14, size = 65, normalized size = 0.86 \[ \frac {a^{2} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-4 a b \arctanh \left ({\mathrm e}^{d x +c}\right )+b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
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 = 0.33, size = 170, normalized size = 2.24 \[ -\frac {1}{8} \, b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
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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mupad [B] time = 0.13, size = 163, normalized size = 2.14 \[ \frac {b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {4\,a^2}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}}-\frac {b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^2\,x}{2}-\frac {8\,a^2}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^3)^2/sinh(c + d*x)^4,x)
[Out]
(b^2*exp(2*c + 2*d*x))/(8*d) - (4*a^2)/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (4*atan((a*b*exp(d*x)
*exp(c)*(-d^2)^(1/2))/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(-d^2)^(1/2) - (b^2*exp(- 2*c - 2*d*x))/(8*d) - (b
^2*x)/2 - (8*a^2)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*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)**4*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
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