3.3.21 \(\int \text {csch}^6(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [221]

Optimal. Leaf size=148 \[ -\frac {1}{16} b^2 (24 a+5 b) x-\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {b^2 (24 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d} \]

[Out]

-1/16*b^2*(24*a+5*b)*x-a^2*(a+3*b)*coth(d*x+c)/d+2/3*a^3*coth(d*x+c)^3/d-1/5*a^3*coth(d*x+c)^5/d+1/16*b^2*(24*
a+11*b)*cosh(d*x+c)*sinh(d*x+c)/d-13/24*b^3*cosh(d*x+c)^3*sinh(d*x+c)/d+1/6*b^3*cosh(d*x+c)^5*sinh(d*x+c)/d

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Rubi [A]
time = 0.26, antiderivative size = 148, 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 = {3296, 1273, 1819, 1816, 213} \begin {gather*} -\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {b^2 (24 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} b^2 x (24 a+5 b)+\frac {b^3 \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b^3 \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/16*(b^2*(24*a + 5*b)*x) - (a^2*(a + 3*b)*Coth[c + d*x])/d + (2*a^3*Coth[c + d*x]^3)/(3*d) - (a^3*Coth[c + d
*x]^5)/(5*d) + (b^2*(24*a + 11*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (13*b^3*Cosh[c + d*x]^3*Sinh[c + d*x])
/(24*d) + (b^3*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*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}^6(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^6 \left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-6 a^3+30 a^3 x^2-6 a^2 (10 a+3 b) x^4+\left (60 a^3+54 a^2 b+b^3\right ) x^6-6 (5 a-b) (a+b)^2 x^8+6 (a+b)^3 x^{10}}{x^6 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {24 a^3-96 a^3 x^2+72 a^2 (2 a+b) x^4-3 \left (32 a^3+48 a^2 b-3 b^3\right ) x^6+24 (a+b)^3 x^8}{x^6 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{24 d}\\ &=\frac {b^2 (24 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-48 a^3+144 a^3 x^2-144 a^2 (a+b) x^4+3 \left (16 a^3+48 a^2 b+24 a b^2+5 b^3\right ) x^6}{x^6 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac {b^2 (24 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \left (-\frac {48 a^3}{x^6}+\frac {96 a^3}{x^4}-\frac {48 a^2 (a+3 b)}{x^2}-\frac {3 b^2 (24 a+5 b)}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {b^2 (24 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {\left (b^2 (24 a+5 b)\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {1}{16} b^2 (24 a+5 b) x-\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {b^2 (24 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^3 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^3 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 110, normalized size = 0.74 \begin {gather*} \frac {-64 a^2 \coth (c+d x) \left (8 a+45 b-4 a \text {csch}^2(c+d x)+3 a \text {csch}^4(c+d x)\right )+5 b^2 (-288 a c-60 b c-288 a d x-60 b d x+9 (16 a+5 b) \sinh (2 (c+d x))-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x)))}{960 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-64*a^2*Coth[c + d*x]*(8*a + 45*b - 4*a*Csch[c + d*x]^2 + 3*a*Csch[c + d*x]^4) + 5*b^2*(-288*a*c - 60*b*c - 2
88*a*d*x - 60*b*d*x + 9*(16*a + 5*b)*Sinh[2*(c + d*x)] - 9*b*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)]))/(960*d)

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Maple [A]
time = 1.55, size = 253, normalized size = 1.71

method	result	size
risch	\(-\frac {3 a \,b^{2} x}{2}-\frac {5 b^{3} x}{16}+\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 b^{3} {\mathrm e}^{4 d x +4 c}}{128 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}+\frac {15 b^{3} {\mathrm e}^{2 d x +2 c}}{128 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}-\frac {15 b^{3} {\mathrm e}^{-2 d x -2 c}}{128 d}+\frac {3 b^{3} {\mathrm e}^{-4 d x -4 c}}{128 d}-\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{384 d}-\frac {2 a^{2} \left (45 b \,{\mathrm e}^{8 d x +8 c}-180 b \,{\mathrm e}^{6 d x +6 c}+80 a \,{\mathrm e}^{4 d x +4 c}+270 b \,{\mathrm e}^{4 d x +4 c}-40 a \,{\mathrm e}^{2 d x +2 c}-180 b \,{\mathrm e}^{2 d x +2 c}+8 a +45 b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^6*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

-3/2*a*b^2*x-5/16*b^3*x+1/384*b^3/d*exp(6*d*x+6*c)-3/128*b^3/d*exp(4*d*x+4*c)+3/8/d*exp(2*d*x+2*c)*a*b^2+15/12
8*b^3/d*exp(2*d*x+2*c)-3/8/d*exp(-2*d*x-2*c)*a*b^2-15/128*b^3/d*exp(-2*d*x-2*c)+3/128*b^3/d*exp(-4*d*x-4*c)-1/
384*b^3/d*exp(-6*d*x-6*c)-2/15*a^2*(45*b*exp(8*d*x+8*c)-180*b*exp(6*d*x+6*c)+80*a*exp(4*d*x+4*c)+270*b*exp(4*d
*x+4*c)-40*a*exp(2*d*x+2*c)-180*b*exp(2*d*x+2*c)+8*a+45*b)/d/(exp(2*d*x+2*c)-1)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (136) = 272\).
time = 0.28, size = 359, normalized size = 2.43 \begin {gather*} -\frac {3}{8} \, a 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 )} - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} - \frac {16}{15} \, a^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-3/8*a*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/384*b^3*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*
c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d)
- 16/15*a^3*(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^2*b/(d*(e^(-2*d*x - 2*c) - 1)
)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (136) = 272\).
time = 0.38, size = 768, normalized size = 5.19 \begin {gather*} \frac {5 \, b^{3} \cosh \left (d x + c\right )^{11} + 55 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 70 \, b^{3} \cosh \left (d x + c\right )^{9} + 15 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{3} - 42 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 20 \, {\left (36 \, a b^{2} + 25 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 70 \, {\left (33 \, b^{3} \cosh \left (d x + c\right )^{5} - 84 \, b^{3} \cosh \left (d x + c\right )^{3} + 2 \, {\left (36 \, a b^{2} + 25 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - {\left (1024 \, a^{3} + 5760 \, a^{2} b + 3600 \, a b^{2} + 1625 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 8 \, {\left (128 \, a^{3} + 720 \, a^{2} b - 15 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (330 \, b^{3} \cosh \left (d x + c\right )^{7} - 1764 \, b^{3} \cosh \left (d x + c\right )^{5} + 140 \, {\left (36 \, a b^{2} + 25 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (1024 \, a^{3} + 5760 \, a^{2} b + 3600 \, a b^{2} + 1625 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 20 \, {\left (256 \, a^{3} + 864 \, a^{2} b + 324 \, a b^{2} + 125 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 40 \, {\left (128 \, a^{3} + 720 \, a^{2} b - 15 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x - 2 \, {\left (128 \, a^{3} + 720 \, a^{2} b - 15 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{9} - 504 \, b^{3} \cosh \left (d x + c\right )^{7} + 84 \, {\left (36 \, a b^{2} + 25 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (1024 \, a^{3} + 5760 \, a^{2} b + 3600 \, a b^{2} + 1625 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (256 \, a^{3} + 864 \, a^{2} b + 324 \, a b^{2} + 125 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 10 \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 360 \, a b^{2} + 131 \, b^{3}\right )} \cosh \left (d x + c\right ) + 40 \, {\left ({\left (128 \, a^{3} + 720 \, a^{2} b - 15 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 256 \, a^{3} + 1440 \, a^{2} b - 30 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x - 3 \, {\left (128 \, a^{3} + 720 \, a^{2} b - 15 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{1920 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, 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)^6*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/1920*(5*b^3*cosh(d*x + c)^11 + 55*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - 70*b^3*cosh(d*x + c)^9 + 15*(55*b^3*c
osh(d*x + c)^3 - 42*b^3*cosh(d*x + c))*sinh(d*x + c)^8 + 20*(36*a*b^2 + 25*b^3)*cosh(d*x + c)^7 + 70*(33*b^3*c
osh(d*x + c)^5 - 84*b^3*cosh(d*x + c)^3 + 2*(36*a*b^2 + 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - (1024*a^3 + 5
760*a^2*b + 3600*a*b^2 + 1625*b^3)*cosh(d*x + c)^5 + 8*(128*a^3 + 720*a^2*b - 15*(24*a*b^2 + 5*b^3)*d*x)*sinh(
d*x + c)^5 + 5*(330*b^3*cosh(d*x + c)^7 - 1764*b^3*cosh(d*x + c)^5 + 140*(36*a*b^2 + 25*b^3)*cosh(d*x + c)^3 -
(1024*a^3 + 5760*a^2*b + 3600*a*b^2 + 1625*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 20*(256*a^3 + 864*a^2*b + 32
4*a*b^2 + 125*b^3)*cosh(d*x + c)^3 - 40*(128*a^3 + 720*a^2*b - 15*(24*a*b^2 + 5*b^3)*d*x - 2*(128*a^3 + 720*a^
2*b - 15*(24*a*b^2 + 5*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 5*(55*b^3*cosh(d*x + c)^9 - 504*b^3*cosh(d
*x + c)^7 + 84*(36*a*b^2 + 25*b^3)*cosh(d*x + c)^5 - 2*(1024*a^3 + 5760*a^2*b + 3600*a*b^2 + 1625*b^3)*cosh(d*
x + c)^3 + 12*(256*a^3 + 864*a^2*b + 324*a*b^2 + 125*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 10*(1024*a^3 + 1152
*a^2*b + 360*a*b^2 + 131*b^3)*cosh(d*x + c) + 40*((128*a^3 + 720*a^2*b - 15*(24*a*b^2 + 5*b^3)*d*x)*cosh(d*x +
c)^4 + 256*a^3 + 1440*a^2*b - 30*(24*a*b^2 + 5*b^3)*d*x - 3*(128*a^3 + 720*a^2*b - 15*(24*a*b^2 + 5*b^3)*d*x)
*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^5 + 5*(2*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^3 + 5*(d*cosh(
d*x + c)^4 - 3*d*cosh(d*x + c)^2 + 2*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)**6*(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. 286 vs. \(2 (136) = 272\).
time = 0.57, size = 286, normalized size = 1.93 \begin {gather*} \frac {5 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 45 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 720 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 225 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 120 \, {\left (24 \, a b^{2} + 5 \, b^{3}\right )} {\left (d x + c\right )} + 5 \, {\left (528 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 110 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 45 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - \frac {256 \, {\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{3} + 45 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{1920 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/1920*(5*b^3*e^(6*d*x + 6*c) - 45*b^3*e^(4*d*x + 4*c) + 720*a*b^2*e^(2*d*x + 2*c) + 225*b^3*e^(2*d*x + 2*c) -
120*(24*a*b^2 + 5*b^3)*(d*x + c) + 5*(528*a*b^2*e^(6*d*x + 6*c) + 110*b^3*e^(6*d*x + 6*c) - 144*a*b^2*e^(4*d*
x + 4*c) - 45*b^3*e^(4*d*x + 4*c) + 9*b^3*e^(2*d*x + 2*c) - b^3)*e^(-6*d*x - 6*c) - 256*(45*a^2*b*e^(8*d*x + 8
*c) - 180*a^2*b*e^(6*d*x + 6*c) + 80*a^3*e^(4*d*x + 4*c) + 270*a^2*b*e^(4*d*x + 4*c) - 40*a^3*e^(2*d*x + 2*c)
- 180*a^2*b*e^(2*d*x + 2*c) + 8*a^3 + 45*a^2*b)/(e^(2*d*x + 2*c) - 1)^5)/d

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Mupad [B]
time = 1.03, size = 511, normalized size = 3.45 \begin {gather*} \frac {\frac {6\,a^2\,b}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{5\,d}+\frac {18\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}-\frac {6\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,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 {\frac {2\,\left (8\,a^3+9\,b\,a^2\right )}{15\,d}-\frac {12\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,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 {6\,a^2\,b}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{5\,d}-\frac {24\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}-\frac {24\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,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 {b^2\,x\,\left (24\,a+5\,b\right )}{16}+\frac {3\,b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{128\,d}-\frac {3\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{128\,d}-\frac {b^3\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{384\,d}+\frac {b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{384\,d}-\frac {3\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (16\,a+5\,b\right )}{128\,d}+\frac {3\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (16\,a+5\,b\right )}{128\,d}-\frac {12\,a^2\,b}{5\,d\,\left ({\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)^6,x)

[Out]

((6*a^2*b)/(5*d) - (2*exp(2*c + 2*d*x)*(9*a^2*b + 8*a^3))/(5*d) + (18*a^2*b*exp(4*c + 4*d*x))/(5*d) - (6*a^2*b
*exp(6*c + 6*d*x))/(5*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
) - ((2*(9*a^2*b + 8*a^3))/(15*d) - (12*a^2*b*exp(2*c + 2*d*x))/(5*d) + (6*a^2*b*exp(4*c + 4*d*x))/(5*d))/(3*e
xp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((6*a^2*b)/(5*d) + (4*exp(4*c + 4*d*x)*(9*a^2*b
+ 8*a^3))/(5*d) - (24*a^2*b*exp(2*c + 2*d*x))/(5*d) - (24*a^2*b*exp(6*c + 6*d*x))/(5*d) + (6*a^2*b*exp(8*c +
8*d*x))/(5*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) - (b^2*x*(24*a + 5*b))/16 + (3*b^3*exp(- 4*c - 4*d*x))/(128*d) - (3*b^3*exp(4*c + 4*d*x))/(128
*d) - (b^3*exp(- 6*c - 6*d*x))/(384*d) + (b^3*exp(6*c + 6*d*x))/(384*d) - (3*b^2*exp(- 2*c - 2*d*x)*(16*a + 5*
b))/(128*d) + (3*b^2*exp(2*c + 2*d*x)*(16*a + 5*b))/(128*d) - (12*a^2*b)/(5*d*(exp(2*c + 2*d*x) - 1))

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