3.3.22 \(\int \text {csch}^8(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [222]

Optimal. Leaf size=133 \[ \frac {3}{8} b^2 (8 a+b) x+\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d} \]

[Out]

3/8*b^2*(8*a+b)*x+a^2*(a+3*b)*coth(d*x+c)/d-a^2*(a+b)*coth(d*x+c)^3/d+3/5*a^3*coth(d*x+c)^5/d-1/7*a^3*coth(d*x
+c)^7/d-5/8*b^3*cosh(d*x+c)*sinh(d*x+c)/d+1/4*b^3*cosh(d*x+c)^3*sinh(d*x+c)/d

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Rubi [A]
time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3296, 1273, 1819, 1816, 213} \begin {gather*} -\frac {a^3 \coth ^7(c+d x)}{7 d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {3}{8} b^2 x (8 a+b)+\frac {b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac {5 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(3*b^2*(8*a + b)*x)/8 + (a^2*(a + 3*b)*Coth[c + d*x])/d - (a^2*(a + b)*Coth[c + d*x]^3)/d + (3*a^3*Coth[c + d*
x]^5)/(5*d) - (a^3*Coth[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]^3*S
inh[c + d*x])/(4*d)

Rule 213

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

Rule 1273

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/(d + e*x^2))*(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))], 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 1816

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 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 3296

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]

Rubi steps

\begin {align*} \int \text {csch}^8(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{x^8 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {4 a^3-20 a^3 x^2+4 a^2 (10 a+3 b) x^4-4 a^2 (10 a+9 b) x^6+\left (20 a^3+36 a^2 b+12 a b^2-b^3\right ) x^8-4 (a+b)^3 x^{10}}{x^8 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {-8 a^3+32 a^3 x^2-24 a^2 (2 a+b) x^4+16 a^2 (2 a+3 b) x^6-\left (8 a^3+24 a^2 b+24 a b^2+3 b^3\right ) x^8}{x^8 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \left (-\frac {8 a^3}{x^8}+\frac {24 a^3}{x^6}-\frac {24 a^2 (a+b)}{x^4}+\frac {8 a^2 (a+3 b)}{x^2}+\frac {3 b^2 (8 a+b)}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\left (3 b^2 (8 a+b)\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {3}{8} b^2 (8 a+b) x+\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 106, normalized size = 0.80 \begin {gather*} \frac {-32 a^2 \coth (c+d x) \left (-2 (8 a+35 b)+(8 a+35 b) \text {csch}^2(c+d x)-6 a \text {csch}^4(c+d x)+5 a \text {csch}^6(c+d x)\right )+35 b^2 (12 (8 a+b) (c+d x)-8 b \sinh (2 (c+d x))+b \sinh (4 (c+d x)))}{1120 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-32*a^2*Coth[c + d*x]*(-2*(8*a + 35*b) + (8*a + 35*b)*Csch[c + d*x]^2 - 6*a*Csch[c + d*x]^4 + 5*a*Csch[c + d*
x]^6) + 35*b^2*(12*(8*a + b)*(c + d*x) - 8*b*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)]))/(1120*d)

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Maple [A]
time = 1.54, size = 207, normalized size = 1.56

method	result	size
risch	\(3 a \,b^{2} x +\frac {3 b^{3} x}{8}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{64 d}-\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{8 d}+\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {4 a^{2} \left (105 b \,{\mathrm e}^{10 d x +10 c}-455 b \,{\mathrm e}^{8 d x +8 c}+280 a \,{\mathrm e}^{6 d x +6 c}+770 b \,{\mathrm e}^{6 d x +6 c}-168 a \,{\mathrm e}^{4 d x +4 c}-630 b \,{\mathrm e}^{4 d x +4 c}+56 a \,{\mathrm e}^{2 d x +2 c}+245 b \,{\mathrm e}^{2 d x +2 c}-8 a -35 b \right )}{35 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{7}}\)	\(207\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

3*a*b^2*x+3/8*b^3*x+1/64*b^3/d*exp(4*d*x+4*c)-1/8*b^3/d*exp(2*d*x+2*c)+1/8*b^3/d*exp(-2*d*x-2*c)-1/64*b^3/d*ex
p(-4*d*x-4*c)-4/35*a^2*(105*b*exp(10*d*x+10*c)-455*b*exp(8*d*x+8*c)+280*a*exp(6*d*x+6*c)+770*b*exp(6*d*x+6*c)-
168*a*exp(4*d*x+4*c)-630*b*exp(4*d*x+4*c)+56*a*exp(2*d*x+2*c)+245*b*exp(2*d*x+2*c)-8*a-35*b)/d/(exp(2*d*x+2*c)
-1)^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (123) = 246\).
time = 0.28, size = 537, normalized size = 4.04 \begin {gather*} \frac {1}{64} \, b^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + 3 \, a b^{2} x + \frac {32}{35} \, a^{3} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {21 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + 4 \, a^{2} b {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/64*b^3*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 3*a*b^
2*x + 32/35*a^3*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^
(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) - 21*e^(-4*d*x - 4*c)
/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10
*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*
d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14
*d*x - 14*c) - 1)) - 1/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c
) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1))) + 4*a^2*b*(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] Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (123) = 246\).
time = 0.42, size = 928, normalized size = 6.98 \begin {gather*} \frac {35 \, b^{3} \cosh \left (d x + c\right )^{11} + 385 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 525 \, b^{3} \cosh \left (d x + c\right )^{9} + 525 \, {\left (11 \, b^{3} \cosh \left (d x + c\right )^{3} - 9 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 8 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (2310 \, b^{3} \cosh \left (d x + c\right )^{5} - 6300 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 7 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x - 3 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (330 \, b^{3} \cosh \left (d x + c\right )^{7} - 1890 \, b^{3} \cosh \left (d x + c\right )^{5} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 42 \, {\left (512 \, a^{3} + 1600 \, a^{2} b + 215 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 56 \, {\left (5 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 384 \, a^{3} + 1680 \, a^{2} b - 315 \, {\left (8 \, a b^{2} + b^{3}\right )} d x - 10 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (275 \, b^{3} \cosh \left (d x + c\right )^{9} - 2700 \, b^{3} \cosh \left (d x + c\right )^{7} + 3 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 18 \, {\left (512 \, a^{3} + 1600 \, a^{2} b + 215 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 70 \, {\left (512 \, a^{3} + 576 \, a^{2} b + 63 \, b^{3}\right )} \cosh \left (d x + c\right ) - 56 \, {\left ({\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{6} - 5 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} - 640 \, a^{3} - 2800 \, a^{2} b + 525 \, {\left (8 \, a b^{2} + b^{3}\right )} d x + 9 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2240 \, {\left (d \sinh \left (d x + c\right )^{7} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{5} + 7 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 10 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (d \cosh \left (d x + c\right )^{6} - 5 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} - 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/2240*(35*b^3*cosh(d*x + c)^11 + 385*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - 525*b^3*cosh(d*x + c)^9 + 525*(11*b
^3*cosh(d*x + c)^3 - 9*b^3*cosh(d*x + c))*sinh(d*x + c)^8 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^7
- 8*(128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x)*sinh(d*x + c)^7 + 7*(2310*b^3*cosh(d*x + c)^5 - 6300*b^3*
cosh(d*x + c)^3 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 7*(1024*a^3 + 4480*a^2*b
+ 975*b^3)*cosh(d*x + c)^5 + 56*(128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x - 3*(128*a^3 + 560*a^2*b - 105
*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*(330*b^3*cosh(d*x + c)^7 - 1890*b^3*cosh(d*x + c)^
5 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^3 - (1024*a^3 + 4480*a^2*b + 975*b^3)*cosh(d*x + c))*sinh
(d*x + c)^4 + 42*(512*a^3 + 1600*a^2*b + 215*b^3)*cosh(d*x + c)^3 - 56*(5*(128*a^3 + 560*a^2*b - 105*(8*a*b^2
+ b^3)*d*x)*cosh(d*x + c)^4 + 384*a^3 + 1680*a^2*b - 315*(8*a*b^2 + b^3)*d*x - 10*(128*a^3 + 560*a^2*b - 105*(
8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(275*b^3*cosh(d*x + c)^9 - 2700*b^3*cosh(d*x + c)^7 +
3*(1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^5 - 10*(1024*a^3 + 4480*a^2*b + 975*b^3)*cosh(d*x + c)^3 +
18*(512*a^3 + 1600*a^2*b + 215*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 70*(512*a^3 + 576*a^2*b + 63*b^3)*cosh(d
*x + c) - 56*((128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^6 - 5*(128*a^3 + 560*a^2*b - 105*(
8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 - 640*a^3 - 2800*a^2*b + 525*(8*a*b^2 + b^3)*d*x + 9*(128*a^3 + 560*a^2*b
- 105*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^7 + 7*(3*d*cosh(d*x + c)^2 - d)*si
nh(d*x + c)^5 + 7*(5*d*cosh(d*x + c)^4 - 10*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^3 + 7*(d*cosh(d*x + c)^6 -
5*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**8*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (123) = 246\).
time = 0.58, size = 253, normalized size = 1.90 \begin {gather*} \frac {35 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 280 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 840 \, {\left (8 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} - 35 \, {\left (144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - \frac {256 \, {\left (105 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 455 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 280 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 770 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 168 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 245 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 35 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}}}{2240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/2240*(35*b^3*e^(4*d*x + 4*c) - 280*b^3*e^(2*d*x + 2*c) + 840*(8*a*b^2 + b^3)*(d*x + c) - 35*(144*a*b^2*e^(4*
d*x + 4*c) + 18*b^3*e^(4*d*x + 4*c) - 8*b^3*e^(2*d*x + 2*c) + b^3)*e^(-4*d*x - 4*c) - 256*(105*a^2*b*e^(10*d*x
+ 10*c) - 455*a^2*b*e^(8*d*x + 8*c) + 280*a^3*e^(6*d*x + 6*c) + 770*a^2*b*e^(6*d*x + 6*c) - 168*a^3*e^(4*d*x
+ 4*c) - 630*a^2*b*e^(4*d*x + 4*c) + 56*a^3*e^(2*d*x + 2*c) + 245*a^2*b*e^(2*d*x + 2*c) - 8*a^3 - 35*a^2*b)/(e
^(2*d*x + 2*c) - 1)^7)/d

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Mupad [B]
time = 1.03, size = 749, normalized size = 5.63 \begin {gather*} \frac {\frac {32\,a^2\,b}{35\,d}-\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{35\,d}+\frac {192\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{35\,d}-\frac {16\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{7\,d}+\frac {24\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}+\frac {24\,a^2\,b\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1}-\frac {\frac {4\,a^2\,b}{7\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{7\,d}-\frac {32\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}-\frac {64\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}+\frac {20\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {\frac {32\,a^2\,b}{35\,d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {4\,\left (8\,a^3+9\,b\,a^2\right )}{35\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{35\,d}+\frac {12\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {3\,b^2\,x\,\left (8\,a+b\right )}{8}+\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d}-\frac {4\,a^2\,b}{7\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^8,x)

[Out]

((32*a^2*b)/(35*d) - (16*exp(2*c + 2*d*x)*(9*a^2*b + 8*a^3))/(35*d) + (192*a^2*b*exp(4*c + 4*d*x))/(35*d) - (1
6*a^2*b*exp(6*c + 6*d*x))/(7*d))/(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c +
8*d*x) + exp(10*c + 10*d*x) - 1) - ((16*exp(6*c + 6*d*x)*(9*a^2*b + 8*a^3))/(7*d) + (24*a^2*b*exp(2*c + 2*d*x
))/(7*d) - (96*a^2*b*exp(4*c + 4*d*x))/(7*d) - (96*a^2*b*exp(8*c + 8*d*x))/(7*d) + (24*a^2*b*exp(10*c + 10*d*x
))/(7*d))/(7*exp(2*c + 2*d*x) - 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) - 35*exp(8*c + 8*d*x) + 21*exp(10*c
+ 10*d*x) - 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) - 1) - ((4*a^2*b)/(7*d) + (8*exp(4*c + 4*d*x)*(9*a^2*b +
8*a^3))/(7*d) - (32*a^2*b*exp(2*c + 2*d*x))/(7*d) - (64*a^2*b*exp(6*c + 6*d*x))/(7*d) + (20*a^2*b*exp(8*c + 8
*d*x))/(7*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10
*c + 10*d*x) + exp(12*c + 12*d*x) + 1) + ((32*a^2*b)/(35*d) - (8*a^2*b*exp(2*c + 2*d*x))/(7*d))/(3*exp(2*c + 2
*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((4*(9*a^2*b + 8*a^3))/(35*d) - (96*a^2*b*exp(2*c + 2*d*x
))/(35*d) + (12*a^2*b*exp(4*c + 4*d*x))/(7*d))/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) +
exp(8*c + 8*d*x) + 1) + (3*b^2*x*(8*a + b))/8 + (b^3*exp(- 2*c - 2*d*x))/(8*d) - (b^3*exp(2*c + 2*d*x))/(8*d)
- (b^3*exp(- 4*c - 4*d*x))/(64*d) + (b^3*exp(4*c + 4*d*x))/(64*d) - (4*a^2*b)/(7*d*(exp(4*c + 4*d*x) - 2*exp(
2*c + 2*d*x) + 1))

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