3.1045 \(\int x^5 \cosh ^7(a+b x^3) \sinh (a+b x^3) \, dx\)

Optimal. Leaf size=129 \[ -\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{192 b^2}-\frac {7 \sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{1152 b^2}-\frac {35 \sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4608 b^2}-\frac {35 \sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{3072 b^2}+\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 x^3}{3072 b} \]

[Out]

-35/3072*x^3/b+1/24*x^3*cosh(b*x^3+a)^8/b-35/3072*cosh(b*x^3+a)*sinh(b*x^3+a)/b^2-35/4608*cosh(b*x^3+a)^3*sinh
(b*x^3+a)/b^2-7/1152*cosh(b*x^3+a)^5*sinh(b*x^3+a)/b^2-1/192*cosh(b*x^3+a)^7*sinh(b*x^3+a)/b^2

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Rubi [A]  time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5373, 5321, 2635, 8} \[ -\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{192 b^2}-\frac {7 \sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{1152 b^2}-\frac {35 \sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4608 b^2}-\frac {35 \sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{3072 b^2}+\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 x^3}{3072 b} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Cosh[a + b*x^3]^7*Sinh[a + b*x^3],x]

[Out]

(-35*x^3)/(3072*b) + (x^3*Cosh[a + b*x^3]^8)/(24*b) - (35*Cosh[a + b*x^3]*Sinh[a + b*x^3])/(3072*b^2) - (35*Co
sh[a + b*x^3]^3*Sinh[a + b*x^3])/(4608*b^2) - (7*Cosh[a + b*x^3]^5*Sinh[a + b*x^3])/(1152*b^2) - (Cosh[a + b*x
^3]^7*Sinh[a + b*x^3])/(192*b^2)

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 5321

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\int x^2 \cosh ^8\left (a+b x^3\right ) \, dx}{8 b}\\ &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\operatorname {Subst}\left (\int \cosh ^8(a+b x) \, dx,x,x^3\right )}{24 b}\\ &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac {7 \operatorname {Subst}\left (\int \cosh ^6(a+b x) \, dx,x,x^3\right )}{192 b}\\ &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac {35 \operatorname {Subst}\left (\int \cosh ^4(a+b x) \, dx,x,x^3\right )}{1152 b}\\ &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac {7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac {35 \operatorname {Subst}\left (\int \cosh ^2(a+b x) \, dx,x,x^3\right )}{1536 b}\\ &=\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 \cosh \left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{3072 b^2}-\frac {35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac {7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac {35 \operatorname {Subst}\left (\int 1 \, dx,x,x^3\right )}{3072 b}\\ &=-\frac {35 x^3}{3072 b}+\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 \cosh \left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{3072 b^2}-\frac {35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac {7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 120, normalized size = 0.93 \[ \frac {-672 \sinh \left (2 \left (a+b x^3\right )\right )-168 \sinh \left (4 \left (a+b x^3\right )\right )-32 \sinh \left (6 \left (a+b x^3\right )\right )-3 \sinh \left (8 \left (a+b x^3\right )\right )+1344 b x^3 \cosh \left (2 \left (a+b x^3\right )\right )+672 b x^3 \cosh \left (4 \left (a+b x^3\right )\right )+192 b x^3 \cosh \left (6 \left (a+b x^3\right )\right )+24 b x^3 \cosh \left (8 \left (a+b x^3\right )\right )}{73728 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Cosh[a + b*x^3]^7*Sinh[a + b*x^3],x]

[Out]

(1344*b*x^3*Cosh[2*(a + b*x^3)] + 672*b*x^3*Cosh[4*(a + b*x^3)] + 192*b*x^3*Cosh[6*(a + b*x^3)] + 24*b*x^3*Cos
h[8*(a + b*x^3)] - 672*Sinh[2*(a + b*x^3)] - 168*Sinh[4*(a + b*x^3)] - 32*Sinh[6*(a + b*x^3)] - 3*Sinh[8*(a +
b*x^3)])/(73728*b^2)

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fricas [B]  time = 0.47, size = 396, normalized size = 3.07 \[ \frac {3 \, b x^{3} \cosh \left (b x^{3} + a\right )^{8} + 3 \, b x^{3} \sinh \left (b x^{3} + a\right )^{8} + 24 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 84 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} - 3 \, \cosh \left (b x^{3} + a\right ) \sinh \left (b x^{3} + a\right )^{7} + 12 \, {\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 2 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{6} + 168 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} - 3 \, {\left (7 \, \cosh \left (b x^{3} + a\right )^{3} + 8 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{5} + 6 \, {\left (35 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 60 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{4} - {\left (21 \, \cosh \left (b x^{3} + a\right )^{5} + 80 \, \cosh \left (b x^{3} + a\right )^{3} + 84 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{3} + 12 \, {\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 30 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 42 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{2} - 3 \, {\left (\cosh \left (b x^{3} + a\right )^{7} + 8 \, \cosh \left (b x^{3} + a\right )^{5} + 28 \, \cosh \left (b x^{3} + a\right )^{3} + 56 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )}{9216 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="fricas")

[Out]

1/9216*(3*b*x^3*cosh(b*x^3 + a)^8 + 3*b*x^3*sinh(b*x^3 + a)^8 + 24*b*x^3*cosh(b*x^3 + a)^6 + 84*b*x^3*cosh(b*x
^3 + a)^4 - 3*cosh(b*x^3 + a)*sinh(b*x^3 + a)^7 + 12*(7*b*x^3*cosh(b*x^3 + a)^2 + 2*b*x^3)*sinh(b*x^3 + a)^6 +
168*b*x^3*cosh(b*x^3 + a)^2 - 3*(7*cosh(b*x^3 + a)^3 + 8*cosh(b*x^3 + a))*sinh(b*x^3 + a)^5 + 6*(35*b*x^3*cos
h(b*x^3 + a)^4 + 60*b*x^3*cosh(b*x^3 + a)^2 + 14*b*x^3)*sinh(b*x^3 + a)^4 - (21*cosh(b*x^3 + a)^5 + 80*cosh(b*
x^3 + a)^3 + 84*cosh(b*x^3 + a))*sinh(b*x^3 + a)^3 + 12*(7*b*x^3*cosh(b*x^3 + a)^6 + 30*b*x^3*cosh(b*x^3 + a)^
4 + 42*b*x^3*cosh(b*x^3 + a)^2 + 14*b*x^3)*sinh(b*x^3 + a)^2 - 3*(cosh(b*x^3 + a)^7 + 8*cosh(b*x^3 + a)^5 + 28
*cosh(b*x^3 + a)^3 + 56*cosh(b*x^3 + a))*sinh(b*x^3 + a))/b^2

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giac [B]  time = 0.14, size = 382, normalized size = 2.96 \[ \frac {24 \, {\left (b x^{3} + a\right )} e^{\left (8 \, b x^{3} + 8 \, a\right )} - 24 \, a e^{\left (8 \, b x^{3} + 8 \, a\right )} + 192 \, {\left (b x^{3} + a\right )} e^{\left (6 \, b x^{3} + 6 \, a\right )} - 192 \, a e^{\left (6 \, b x^{3} + 6 \, a\right )} + 672 \, {\left (b x^{3} + a\right )} e^{\left (4 \, b x^{3} + 4 \, a\right )} - 672 \, a e^{\left (4 \, b x^{3} + 4 \, a\right )} + 1344 \, {\left (b x^{3} + a\right )} e^{\left (2 \, b x^{3} + 2 \, a\right )} - 1344 \, a e^{\left (2 \, b x^{3} + 2 \, a\right )} + 1344 \, {\left (b x^{3} + a\right )} e^{\left (-2 \, b x^{3} - 2 \, a\right )} - 1344 \, a e^{\left (-2 \, b x^{3} - 2 \, a\right )} + 672 \, {\left (b x^{3} + a\right )} e^{\left (-4 \, b x^{3} - 4 \, a\right )} - 672 \, a e^{\left (-4 \, b x^{3} - 4 \, a\right )} + 192 \, {\left (b x^{3} + a\right )} e^{\left (-6 \, b x^{3} - 6 \, a\right )} - 192 \, a e^{\left (-6 \, b x^{3} - 6 \, a\right )} + 24 \, {\left (b x^{3} + a\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )} - 24 \, a e^{\left (-8 \, b x^{3} - 8 \, a\right )} - 3 \, e^{\left (8 \, b x^{3} + 8 \, a\right )} - 32 \, e^{\left (6 \, b x^{3} + 6 \, a\right )} - 168 \, e^{\left (4 \, b x^{3} + 4 \, a\right )} - 672 \, e^{\left (2 \, b x^{3} + 2 \, a\right )} + 672 \, e^{\left (-2 \, b x^{3} - 2 \, a\right )} + 168 \, e^{\left (-4 \, b x^{3} - 4 \, a\right )} + 32 \, e^{\left (-6 \, b x^{3} - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{147456 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="giac")

[Out]

1/147456*(24*(b*x^3 + a)*e^(8*b*x^3 + 8*a) - 24*a*e^(8*b*x^3 + 8*a) + 192*(b*x^3 + a)*e^(6*b*x^3 + 6*a) - 192*
a*e^(6*b*x^3 + 6*a) + 672*(b*x^3 + a)*e^(4*b*x^3 + 4*a) - 672*a*e^(4*b*x^3 + 4*a) + 1344*(b*x^3 + a)*e^(2*b*x^
3 + 2*a) - 1344*a*e^(2*b*x^3 + 2*a) + 1344*(b*x^3 + a)*e^(-2*b*x^3 - 2*a) - 1344*a*e^(-2*b*x^3 - 2*a) + 672*(b
*x^3 + a)*e^(-4*b*x^3 - 4*a) - 672*a*e^(-4*b*x^3 - 4*a) + 192*(b*x^3 + a)*e^(-6*b*x^3 - 6*a) - 192*a*e^(-6*b*x
^3 - 6*a) + 24*(b*x^3 + a)*e^(-8*b*x^3 - 8*a) - 24*a*e^(-8*b*x^3 - 8*a) - 3*e^(8*b*x^3 + 8*a) - 32*e^(6*b*x^3
+ 6*a) - 168*e^(4*b*x^3 + 4*a) - 672*e^(2*b*x^3 + 2*a) + 672*e^(-2*b*x^3 - 2*a) + 168*e^(-4*b*x^3 - 4*a) + 32*
e^(-6*b*x^3 - 6*a) + 3*e^(-8*b*x^3 - 8*a))/b^2

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maple [A]  time = 0.37, size = 194, normalized size = 1.50 \[ \frac {\left (8 b \,x^{3}-1\right ) {\mathrm e}^{8 b \,x^{3}+8 a}}{49152 b^{2}}+\frac {\left (6 b \,x^{3}-1\right ) {\mathrm e}^{6 b \,x^{3}+6 a}}{4608 b^{2}}+\frac {7 \left (4 b \,x^{3}-1\right ) {\mathrm e}^{4 b \,x^{3}+4 a}}{6144 b^{2}}+\frac {7 \left (2 b \,x^{3}-1\right ) {\mathrm e}^{2 b \,x^{3}+2 a}}{1536 b^{2}}+\frac {7 \left (2 b \,x^{3}+1\right ) {\mathrm e}^{-2 b \,x^{3}-2 a}}{1536 b^{2}}+\frac {7 \left (4 b \,x^{3}+1\right ) {\mathrm e}^{-4 b \,x^{3}-4 a}}{6144 b^{2}}+\frac {\left (6 b \,x^{3}+1\right ) {\mathrm e}^{-6 b \,x^{3}-6 a}}{4608 b^{2}}+\frac {\left (8 b \,x^{3}+1\right ) {\mathrm e}^{-8 b \,x^{3}-8 a}}{49152 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x)

[Out]

1/49152*(8*b*x^3-1)/b^2*exp(8*b*x^3+8*a)+1/4608*(6*b*x^3-1)/b^2*exp(6*b*x^3+6*a)+7/6144*(4*b*x^3-1)/b^2*exp(4*
b*x^3+4*a)+7/1536*(2*b*x^3-1)/b^2*exp(2*b*x^3+2*a)+7/1536*(2*b*x^3+1)/b^2*exp(-2*b*x^3-2*a)+7/6144*(4*b*x^3+1)
/b^2*exp(-4*b*x^3-4*a)+1/4608*(6*b*x^3+1)/b^2*exp(-6*b*x^3-6*a)+1/49152*(8*b*x^3+1)/b^2*exp(-8*b*x^3-8*a)

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maxima [A]  time = 0.34, size = 213, normalized size = 1.65 \[ \frac {{\left (8 \, b x^{3} e^{\left (8 \, a\right )} - e^{\left (8 \, a\right )}\right )} e^{\left (8 \, b x^{3}\right )}}{49152 \, b^{2}} + \frac {{\left (6 \, b x^{3} e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x^{3}\right )}}{4608 \, b^{2}} + \frac {7 \, {\left (4 \, b x^{3} e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x^{3}\right )}}{6144 \, b^{2}} + \frac {7 \, {\left (2 \, b x^{3} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{3}\right )}}{1536 \, b^{2}} + \frac {7 \, {\left (2 \, b x^{3} + 1\right )} e^{\left (-2 \, b x^{3} - 2 \, a\right )}}{1536 \, b^{2}} + \frac {7 \, {\left (4 \, b x^{3} + 1\right )} e^{\left (-4 \, b x^{3} - 4 \, a\right )}}{6144 \, b^{2}} + \frac {{\left (6 \, b x^{3} + 1\right )} e^{\left (-6 \, b x^{3} - 6 \, a\right )}}{4608 \, b^{2}} + \frac {{\left (8 \, b x^{3} + 1\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{49152 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="maxima")

[Out]

1/49152*(8*b*x^3*e^(8*a) - e^(8*a))*e^(8*b*x^3)/b^2 + 1/4608*(6*b*x^3*e^(6*a) - e^(6*a))*e^(6*b*x^3)/b^2 + 7/6
144*(4*b*x^3*e^(4*a) - e^(4*a))*e^(4*b*x^3)/b^2 + 7/1536*(2*b*x^3*e^(2*a) - e^(2*a))*e^(2*b*x^3)/b^2 + 7/1536*
(2*b*x^3 + 1)*e^(-2*b*x^3 - 2*a)/b^2 + 7/6144*(4*b*x^3 + 1)*e^(-4*b*x^3 - 4*a)/b^2 + 1/4608*(6*b*x^3 + 1)*e^(-
6*b*x^3 - 6*a)/b^2 + 1/49152*(8*b*x^3 + 1)*e^(-8*b*x^3 - 8*a)/b^2

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mupad [B]  time = 0.28, size = 213, normalized size = 1.65 \[ {\mathrm {e}}^{-2\,b\,x^3-2\,a}\,\left (\frac {7}{1536\,b^2}+\frac {7\,x^3}{768\,b}\right )-{\mathrm {e}}^{2\,b\,x^3+2\,a}\,\left (\frac {7}{1536\,b^2}-\frac {7\,x^3}{768\,b}\right )+{\mathrm {e}}^{-6\,b\,x^3-6\,a}\,\left (\frac {1}{4608\,b^2}+\frac {x^3}{768\,b}\right )-{\mathrm {e}}^{6\,b\,x^3+6\,a}\,\left (\frac {1}{4608\,b^2}-\frac {x^3}{768\,b}\right )+{\mathrm {e}}^{-4\,b\,x^3-4\,a}\,\left (\frac {7}{6144\,b^2}+\frac {7\,x^3}{1536\,b}\right )-{\mathrm {e}}^{4\,b\,x^3+4\,a}\,\left (\frac {7}{6144\,b^2}-\frac {7\,x^3}{1536\,b}\right )+{\mathrm {e}}^{-8\,b\,x^3-8\,a}\,\left (\frac {1}{49152\,b^2}+\frac {x^3}{6144\,b}\right )-{\mathrm {e}}^{8\,b\,x^3+8\,a}\,\left (\frac {1}{49152\,b^2}-\frac {x^3}{6144\,b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cosh(a + b*x^3)^7*sinh(a + b*x^3),x)

[Out]

exp(- 2*a - 2*b*x^3)*(7/(1536*b^2) + (7*x^3)/(768*b)) - exp(2*a + 2*b*x^3)*(7/(1536*b^2) - (7*x^3)/(768*b)) +
exp(- 6*a - 6*b*x^3)*(1/(4608*b^2) + x^3/(768*b)) - exp(6*a + 6*b*x^3)*(1/(4608*b^2) - x^3/(768*b)) + exp(- 4*
a - 4*b*x^3)*(7/(6144*b^2) + (7*x^3)/(1536*b)) - exp(4*a + 4*b*x^3)*(7/(6144*b^2) - (7*x^3)/(1536*b)) + exp(-
8*a - 8*b*x^3)*(1/(49152*b^2) + x^3/(6144*b)) - exp(8*a + 8*b*x^3)*(1/(49152*b^2) - x^3/(6144*b))

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sympy [A]  time = 68.21, size = 241, normalized size = 1.87 \[ \begin {cases} - \frac {35 x^{3} \sinh ^{8}{\left (a + b x^{3} \right )}}{3072 b} + \frac {35 x^{3} \sinh ^{6}{\left (a + b x^{3} \right )} \cosh ^{2}{\left (a + b x^{3} \right )}}{768 b} - \frac {35 x^{3} \sinh ^{4}{\left (a + b x^{3} \right )} \cosh ^{4}{\left (a + b x^{3} \right )}}{512 b} + \frac {35 x^{3} \sinh ^{2}{\left (a + b x^{3} \right )} \cosh ^{6}{\left (a + b x^{3} \right )}}{768 b} + \frac {31 x^{3} \cosh ^{8}{\left (a + b x^{3} \right )}}{1024 b} + \frac {35 \sinh ^{7}{\left (a + b x^{3} \right )} \cosh {\left (a + b x^{3} \right )}}{3072 b^{2}} - \frac {385 \sinh ^{5}{\left (a + b x^{3} \right )} \cosh ^{3}{\left (a + b x^{3} \right )}}{9216 b^{2}} + \frac {511 \sinh ^{3}{\left (a + b x^{3} \right )} \cosh ^{5}{\left (a + b x^{3} \right )}}{9216 b^{2}} - \frac {31 \sinh {\left (a + b x^{3} \right )} \cosh ^{7}{\left (a + b x^{3} \right )}}{1024 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{6} \sinh {\relax (a )} \cosh ^{7}{\relax (a )}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*cosh(b*x**3+a)**7*sinh(b*x**3+a),x)

[Out]

Piecewise((-35*x**3*sinh(a + b*x**3)**8/(3072*b) + 35*x**3*sinh(a + b*x**3)**6*cosh(a + b*x**3)**2/(768*b) - 3
5*x**3*sinh(a + b*x**3)**4*cosh(a + b*x**3)**4/(512*b) + 35*x**3*sinh(a + b*x**3)**2*cosh(a + b*x**3)**6/(768*
b) + 31*x**3*cosh(a + b*x**3)**8/(1024*b) + 35*sinh(a + b*x**3)**7*cosh(a + b*x**3)/(3072*b**2) - 385*sinh(a +
b*x**3)**5*cosh(a + b*x**3)**3/(9216*b**2) + 511*sinh(a + b*x**3)**3*cosh(a + b*x**3)**5/(9216*b**2) - 31*sin
h(a + b*x**3)*cosh(a + b*x**3)**7/(1024*b**2), Ne(b, 0)), (x**6*sinh(a)*cosh(a)**7/6, True))

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