3.26 \(\int \text {csch}^3(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=83 \[ -\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b^2 (3 a-b) \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d} \]

[Out]

1/2*a^2*(a-6*b)*arctanh(cosh(d*x+c))/d+(3*a-b)*b^2*cosh(d*x+c)/d+1/3*b^3*cosh(d*x+c)^3/d-1/2*a^3*coth(d*x+c)*c
sch(d*x+c)/d

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Rubi [A]  time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3186, 390, 385, 206} \[ \frac {a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b^2 (3 a-b) \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(a^2*(a - 6*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + ((3*a - b)*b^2*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) -
(a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-b+b x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((3 a-b) b^2+b^3 x^2+\frac {a^2 (a-3 b)+3 a^2 b x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(3 a-b) b^2 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-3 b)+3 a^2 b x^2}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(3 a-b) b^2 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {\left (a^2 (a-6 b)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {(3 a-b) b^2 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}

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Mathematica [B]  time = 4.59, size = 210, normalized size = 2.53 \[ -\frac {\left (a+b \sinh ^2(c+d x)\right )^3 \left (3 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+3 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+12 a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-12 a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-72 a^2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+72 a^2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-72 a b^2 \sinh (c) \sinh (d x)-18 b^2 (4 a-b) \cosh (c) \cosh (d x)+18 b^3 \sinh (c) \sinh (d x)-2 b^3 \sinh (3 c) \sinh (3 d x)-2 b^3 \cosh (3 c) \cosh (3 d x)\right )}{3 d (2 a+b \cosh (2 (c+d x))-b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

-1/3*((-18*(4*a - b)*b^2*Cosh[c]*Cosh[d*x] - 2*b^3*Cosh[3*c]*Cosh[3*d*x] + 3*a^3*Csch[(c + d*x)/2]^2 - 12*a^3*
Log[Cosh[(c + d*x)/2]] + 72*a^2*b*Log[Cosh[(c + d*x)/2]] + 12*a^3*Log[Sinh[(c + d*x)/2]] - 72*a^2*b*Log[Sinh[(
c + d*x)/2]] + 3*a^3*Sech[(c + d*x)/2]^2 - 72*a*b^2*Sinh[c]*Sinh[d*x] + 18*b^3*Sinh[c]*Sinh[d*x] - 2*b^3*Sinh[
3*c]*Sinh[3*d*x])*(a + b*Sinh[c + d*x]^2)^3)/(d*(2*a - b + b*Cosh[2*(c + d*x)])^3)

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fricas [B]  time = 0.56, size = 1814, normalized size = 21.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/24*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 + (36*a*b^2 - 11*b^3)
*cosh(d*x + c)^8 + (45*b^3*cosh(d*x + c)^2 + 36*a*b^2 - 11*b^3)*sinh(d*x + c)^8 + 8*(15*b^3*cosh(d*x + c)^3 +
(36*a*b^2 - 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c)^6 + 2*(105*b^
3*cosh(d*x + c)^4 - 12*a^3 - 18*a*b^2 + 5*b^3 + 14*(36*a*b^2 - 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(6
3*b^3*cosh(d*x + c)^5 + 14*(36*a*b^2 - 11*b^3)*cosh(d*x + c)^3 - 3*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c))*
sinh(d*x + c)^5 - 2*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 2*(105*b^3*cosh(d*x + c)^6 + 35*(36*a*b^2 -
11*b^3)*cosh(d*x + c)^4 - 12*a^3 - 18*a*b^2 + 5*b^3 - 15*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 8*(15*b^3*cosh(d*x + c)^7 + 7*(36*a*b^2 - 11*b^3)*cosh(d*x + c)^5 - 5*(12*a^3 + 18*a*b^2 - 5*b^3)*co
sh(d*x + c)^3 - (12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + b^3 + (36*a*b^2 - 11*b^3)*cosh(d*
x + c)^2 + (45*b^3*cosh(d*x + c)^8 + 28*(36*a*b^2 - 11*b^3)*cosh(d*x + c)^6 - 30*(12*a^3 + 18*a*b^2 - 5*b^3)*c
osh(d*x + c)^4 + 36*a*b^2 - 11*b^3 - 12*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((a^
3 - 6*a^2*b)*cosh(d*x + c)^7 + 7*(a^3 - 6*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (a^3 - 6*a^2*b)*sinh(d*x + c)
^7 - 2*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 - (2*a^3 - 12*a^2*b - 21*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)
^5 + 5*(7*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 - 2*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^4 + (a^3 - 6*a^2*b)
*cosh(d*x + c)^3 + (35*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + a^3 - 6*a^2*b - 20*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*s
inh(d*x + c)^3 + (21*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 - 20*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 3*(a^3 - 6*a^2*b)*
cosh(d*x + c))*sinh(d*x + c)^2 + (7*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 10*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + 3*(
a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 12*((a^3 - 6*a^2*b)*co
sh(d*x + c)^7 + 7*(a^3 - 6*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (a^3 - 6*a^2*b)*sinh(d*x + c)^7 - 2*(a^3 - 6
*a^2*b)*cosh(d*x + c)^5 - (2*a^3 - 12*a^2*b - 21*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(a^3
- 6*a^2*b)*cosh(d*x + c)^3 - 2*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^4 + (a^3 - 6*a^2*b)*cosh(d*x + c)^
3 + (35*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + a^3 - 6*a^2*b - 20*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^3
+ (21*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 - 20*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 3*(a^3 - 6*a^2*b)*cosh(d*x + c))*
sinh(d*x + c)^2 + (7*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 10*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + 3*(a^3 - 6*a^2*b)*
cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(5*b^3*cosh(d*x + c)^9 + 4*(36*a*b^
2 - 11*b^3)*cosh(d*x + c)^7 - 6*(12*a^3 + 18*a*b^2 - 5*b^3)*cosh(d*x + c)^5 - 4*(12*a^3 + 18*a*b^2 - 5*b^3)*co
sh(d*x + c)^3 + (36*a*b^2 - 11*b^3)*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*c
osh(d*x + c)^3 - 2*d*cosh(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 = 0.23, size = 174, normalized size = 2.10 \[ \frac {b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 36 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 12 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {24 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4} + 6 \, {\left (a^{3} - 6 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 6 \, {\left (a^{3} - 6 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/24*(b^3*(e^(d*x + c) + e^(-d*x - c))^3 + 36*a*b^2*(e^(d*x + c) + e^(-d*x - c)) - 12*b^3*(e^(d*x + c) + e^(-d
*x - c)) - 24*a^3*(e^(d*x + c) + e^(-d*x - c))/((e^(d*x + c) + e^(-d*x - c))^2 - 4) + 6*(a^3 - 6*a^2*b)*log(e^
(d*x + c) + e^(-d*x - c) + 2) - 6*(a^3 - 6*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) - 2))/d

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maple [A]  time = 0.09, size = 79, normalized size = 0.95 \[ \frac {a^{3} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )-6 a^{2} b \arctanh \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \cosh \left (d x +c \right )+b^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))-6*a^2*b*arctanh(exp(d*x+c))+3*a*b^2*cosh(d*x+c)+b^
3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))

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maxima [B]  time = 0.34, size = 217, normalized size = 2.61 \[ \frac {1}{24} \, b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/24*b^3*(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) + 3/2*a*b^2*(e^(d*x + c
)/d + e^(-d*x - c)/d) + 1/2*a^3*(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))) - 3*a^2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x -
c) - 1)/d)

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mupad [B]  time = 0.22, size = 229, normalized size = 2.76 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {-d^2}-6\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6-12\,a^5\,b+36\,a^4\,b^2}}\right )\,\sqrt {a^6-12\,a^5\,b+36\,a^4\,b^2}}{\sqrt {-d^2}}+\frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}+\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a-b\right )}{8\,d}+\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-b\right )}{8\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^2)^3/sinh(c + d*x)^3,x)

[Out]

(atan((exp(d*x)*exp(c)*(a^3*(-d^2)^(1/2) - 6*a^2*b*(-d^2)^(1/2)))/(d*(a^6 - 12*a^5*b + 36*a^4*b^2)^(1/2)))*(a^
6 - 12*a^5*b + 36*a^4*b^2)^(1/2))/(-d^2)^(1/2) + (b^3*exp(- 3*c - 3*d*x))/(24*d) + (b^3*exp(3*c + 3*d*x))/(24*
d) + (3*b^2*exp(c + d*x)*(4*a - b))/(8*d) + (3*b^2*exp(- c - d*x)*(4*a - b))/(8*d) - (a^3*exp(c + d*x))/(d*(ex
p(2*c + 2*d*x) - 1)) - (2*a^3*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)**2)**3,x)

[Out]

Timed out

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