3.223 \(\int \text{csch}^{10}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=140 \[ -\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac{2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac{a^3 \coth ^9(c+d x)}{9 d}+\frac{4 a^3 \coth ^7(c+d x)}{7 d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2} \]

[Out]

-(b^3*x)/2 - (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d + (2*a^2*(2*a + 3*b)*Coth[c + d*x]^3)/(3*d) - (3*a^2*(2
*a + b)*Coth[c + d*x]^5)/(5*d) + (4*a^3*Coth[c + d*x]^7)/(7*d) - (a^3*Coth[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d
*x]*Sinh[c + d*x])/(2*d)

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Rubi [A]  time = 0.222652, antiderivative size = 140, normalized size of antiderivative = 1., 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 = {3217, 1259, 1802, 207} \[ -\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac{2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac{a^3 \coth ^9(c+d x)}{9 d}+\frac{4 a^3 \coth ^7(c+d x)}{7 d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-(b^3*x)/2 - (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d + (2*a^2*(2*a + 3*b)*Coth[c + d*x]^3)/(3*d) - (3*a^2*(2
*a + b)*Coth[c + d*x]^5)/(5*d) + (4*a^3*Coth[c + d*x]^7)/(7*d) - (a^3*Coth[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d
*x]*Sinh[c + d*x])/(2*d)

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 207

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

Rubi steps

\begin{align*} \int \text{csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-2 a x^2+(a+b) x^4\right )^3}{x^{10} \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{-2 a^3+10 a^3 x^2-2 a^2 (10 a+3 b) x^4+2 a^2 (10 a+9 b) x^6-2 a \left (5 a^2+9 a b+3 b^2\right ) x^8+\left (2 a^3+6 a^2 b+6 a b^2+b^3\right ) x^{10}}{x^{10} \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a^3}{x^{10}}+\frac{8 a^3}{x^8}-\frac{6 a^2 (2 a+b)}{x^6}+\frac{4 a^2 (2 a+3 b)}{x^4}-\frac{2 a \left (a^2+3 a b+3 b^2\right )}{x^2}-\frac{b^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac{2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac{3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac{4 a^3 \coth ^7(c+d x)}{7 d}-\frac{a^3 \coth ^9(c+d x)}{9 d}+\frac{b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac{b^3 x}{2}-\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac{2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac{3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac{4 a^3 \coth ^7(c+d x)}{7 d}-\frac{a^3 \coth ^9(c+d x)}{9 d}+\frac{b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.579417, size = 115, normalized size = 0.82 \[ \frac{315 b^3 (\sinh (2 (c+d x))-2 (c+d x))-4 a \coth (c+d x) \left (35 a^2 \text{csch}^8(c+d x)-40 a^2 \text{csch}^6(c+d x)+128 a^2+3 a (16 a+63 b) \text{csch}^4(c+d x)-4 a (16 a+63 b) \text{csch}^2(c+d x)+504 a b+945 b^2\right )}{1260 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-4*a*Coth[c + d*x]*(128*a^2 + 504*a*b + 945*b^2 - 4*a*(16*a + 63*b)*Csch[c + d*x]^2 + 3*a*(16*a + 63*b)*Csch[
c + d*x]^4 - 40*a^2*Csch[c + d*x]^6 + 35*a^2*Csch[c + d*x]^8) + 315*b^3*(-2*(c + d*x) + Sinh[2*(c + d*x)]))/(1
260*d)

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Maple [A]  time = 0.084, size = 130, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{128}{315}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{9}}+{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{63}}-{\frac{16\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{315}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -{\frac{8}{15}}-1/5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right )-3\,a{b}^{2}{\rm coth} \left (dx+c\right )+{b}^{3} \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)^10*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(a^3*(-128/315-1/9*csch(d*x+c)^8+8/63*csch(d*x+c)^6-16/105*csch(d*x+c)^4+64/315*csch(d*x+c)^2)*coth(d*x+c)
+3*a^2*b*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)-3*a*b^2*coth(d*x+c)+b^3*(1/2*cosh(d*x+c)*sin
h(d*x+c)-1/2*d*x-1/2*c))

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Maxima [B]  time = 1.21684, size = 1137, normalized size = 8.12 \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)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*b^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 256/315*a^3*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*
c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x
- 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 36*e^(-4*d*x - 4*c)/(d*(9
*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c)
- 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) + 84*e^(-6*d
*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-
10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1
)) - 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x -
8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18
*d*x - 18*c) - 1)) - 1/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*
c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*
x - 18*c) - 1))) - 16/5*a^2*b*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x
- 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4
*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d*(5*e^(-2*d*x - 2*c) -
10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1))) + 6*a*b^2/(d*(e^(-
2*d*x - 2*c) - 1))

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Fricas [B]  time = 1.56726, size = 3646, normalized size = 26.04 \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)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/2520*(315*b^3*cosh(d*x + c)^11 + 3465*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - (1024*a^3 + 4032*a^2*b + 7560*a*b
^2 + 2835*b^3)*cosh(d*x + c)^9 - 4*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*sinh(d*x + c)^9 + 9*(5775
*b^3*cosh(d*x + c)^3 - (1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 9*(102
4*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)*cosh(d*x + c)^7 + 36*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*
b^2 - 4*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(6930*b^3*cosh
(d*x + c)^5 - 4*(1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^3 + 3*(1024*a^3 + 4032*a^2*b + 5
880*a*b^2 + 1225*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 9*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 + 2625*b^3)*cos
h(d*x + c)^5 - 36*(1260*b^3*d*x + 14*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^4 - 1024*
a^3 - 4032*a^2*b - 7560*a*b^2 - 21*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x
+ c)^5 + 9*(11550*b^3*cosh(d*x + c)^7 - 14*(1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^5 +
35*(1024*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)*cosh(d*x + c)^3 - 5*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 +
2625*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 42*(2048*a^3 + 6144*a^2*b + 5040*a*b^2 + 675*b^3)*cosh(d*x + c)^3
- 12*(28*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^6 - 8820*b^3*d*x - 105*(315*b^3*d*x -
256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^4 + 7168*a^3 + 28224*a^2*b + 52920*a*b^2 + 120*(315*b^3*d*x
- 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(1925*b^3*cosh(d*x + c)^9 - 4*(1024*
a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^7 + 21*(1024*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)
*cosh(d*x + c)^5 - 10*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 + 2625*b^3)*cosh(d*x + c)^3 + 14*(2048*a^3 + 6144*
a^2*b + 5040*a*b^2 + 675*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 126*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 105*b^
3)*cosh(d*x + c) - 36*((315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^8 - 7*(315*b^3*d*x - 25
6*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^6 + 4410*b^3*d*x + 20*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 189
0*a*b^2)*cosh(d*x + c)^4 - 3584*a^3 - 14112*a^2*b - 26460*a*b^2 - 28*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 189
0*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^9 + 9*(4*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^7 + 9*
(14*d*cosh(d*x + c)^4 - 21*d*cosh(d*x + c)^2 + 4*d)*sinh(d*x + c)^5 + 3*(28*d*cosh(d*x + c)^6 - 105*d*cosh(d*x
+ c)^4 + 120*d*cosh(d*x + c)^2 - 28*d)*sinh(d*x + c)^3 + 9*(d*cosh(d*x + c)^8 - 7*d*cosh(d*x + c)^6 + 20*d*co
sh(d*x + c)^4 - 28*d*cosh(d*x + c)^2 + 14*d)*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)**10*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.71514, size = 495, normalized size = 3.54 \begin{align*} -\frac{{\left (d x + c\right )} b^{3}}{2 \, d} + \frac{b^{3} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac{{\left (2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac{2 \,{\left (945 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 7560 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 26460 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 22680 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 52920 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 16128 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40824 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 66150 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 10752 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 37296 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 52920 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4608 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 26460 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1152 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4536 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 7560 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 128 \, a^{3} + 504 \, a^{2} b + 945 \, a b^{2}\right )}}{315 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/2*(d*x + c)*b^3/d + 1/8*b^3*e^(2*d*x + 2*c)/d + 1/8*(2*b^3*e^(2*d*x + 2*c) - b^3)*e^(-2*d*x - 2*c)/d - 2/31
5*(945*a*b^2*e^(16*d*x + 16*c) - 7560*a*b^2*e^(14*d*x + 14*c) + 5040*a^2*b*e^(12*d*x + 12*c) + 26460*a*b^2*e^(
12*d*x + 12*c) - 22680*a^2*b*e^(10*d*x + 10*c) - 52920*a*b^2*e^(10*d*x + 10*c) + 16128*a^3*e^(8*d*x + 8*c) + 4
0824*a^2*b*e^(8*d*x + 8*c) + 66150*a*b^2*e^(8*d*x + 8*c) - 10752*a^3*e^(6*d*x + 6*c) - 37296*a^2*b*e^(6*d*x +
6*c) - 52920*a*b^2*e^(6*d*x + 6*c) + 4608*a^3*e^(4*d*x + 4*c) + 18144*a^2*b*e^(4*d*x + 4*c) + 26460*a*b^2*e^(4
*d*x + 4*c) - 1152*a^3*e^(2*d*x + 2*c) - 4536*a^2*b*e^(2*d*x + 2*c) - 7560*a*b^2*e^(2*d*x + 2*c) + 128*a^3 + 5
04*a^2*b + 945*a*b^2)/(d*(e^(2*d*x + 2*c) - 1)^9)