3.10 \(\int \sinh ^4(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=146 \[ \frac{\left (48 a^2-208 a b+139 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{\left (80 a^2-176 a b+93 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (48 a^2-80 a b+35 b^2\right )+\frac{b (16 a-13 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b^2 \sinh ^5(c+d x) \cosh ^3(c+d x)}{8 d} \]

[Out]

((48*a^2 - 80*a*b + 35*b^2)*x)/128 - ((80*a^2 - 176*a*b + 93*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*
a^2 - 208*a*b + 139*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((16*a - 13*b)*b*Cosh[c + d*x]^5*Sinh[c + d*
x])/(48*d) + (b^2*Cosh[c + d*x]^3*Sinh[c + d*x]^5)/(8*d)

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Rubi [A]  time = 0.1804, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3187, 463, 455, 1157, 385, 206} \[ \frac{\left (48 a^2-208 a b+139 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{\left (80 a^2-176 a b+93 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (48 a^2-80 a b+35 b^2\right )+\frac{b (16 a-13 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b^2 \sinh ^5(c+d x) \cosh ^3(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((48*a^2 - 80*a*b + 35*b^2)*x)/128 - ((80*a^2 - 176*a*b + 93*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*
a^2 - 208*a*b + 139*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((16*a - 13*b)*b*Cosh[c + d*x]^5*Sinh[c + d*
x])/(48*d) + (b^2*Cosh[c + d*x]^3*Sinh[c + d*x]^5)/(8*d)

Rule 3187

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

Rule 463

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

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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 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])

Rubi steps

\begin{align*} \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a-(a-b) x^2\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (-8 a^2+5 b^2+8 (a-b)^2 x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{(16 a-13 b) b+6 (16 a-13 b) b x^2-48 (a-b)^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac{\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}+\frac{\operatorname{Subst}\left (\int \frac{-3 \left (16 a^2-48 a b+29 b^2\right )-192 (a-b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{192 d}\\ &=-\frac{\left (80 a^2-176 a b+93 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}+\frac{\left (48 a^2-80 a b+35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{1}{128} \left (48 a^2-80 a b+35 b^2\right ) x-\frac{\left (80 a^2-176 a b+93 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.203947, size = 133, normalized size = 0.91 \[ \frac{-96 \left (8 a^2-15 a b+7 b^2\right ) \sinh (2 (c+d x))+24 \left (4 a^2-12 a b+7 b^2\right ) \sinh (4 (c+d x))+1152 a^2 c+1152 a^2 d x+32 a b \sinh (6 (c+d x))-1920 a b c-1920 a b d x-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))+840 b^2 c+840 b^2 d x}{3072 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(1152*a^2*c - 1920*a*b*c + 840*b^2*c + 1152*a^2*d*x - 1920*a*b*d*x + 840*b^2*d*x - 96*(8*a^2 - 15*a*b + 7*b^2)
*Sinh[2*(c + d*x)] + 24*(4*a^2 - 12*a*b + 7*b^2)*Sinh[4*(c + d*x)] + 32*a*b*Sinh[6*(c + d*x)] - 32*b^2*Sinh[6*
(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)

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Maple [A]  time = 0.048, size = 150, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{8}}-{\frac{7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{48}}+{\frac{35\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{192}}-{\frac{35\,\sinh \left ( dx+c \right ) }{128}} \right ) \cosh \left ( dx+c \right ) +{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) +2\,ab \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \right ) +{a}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/d*(b^2*((1/8*sinh(d*x+c)^7-7/48*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*sinh(d*x+c))*cosh(d*x+c)+35/128*d*
x+35/128*c)+2*a*b*((1/6*sinh(d*x+c)^5-5/24*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*c)+a^2*((
1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.04378, size = 360, normalized size = 2.47 \begin{align*} \frac{1}{64} \, a^{2}{\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 )} - \frac{1}{6144} \, b^{2}{\left (\frac{{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{1680 \,{\left (d x + c\right )}}{d} - \frac{672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac{1}{192} \, a b{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/64*a^2*(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) - 1/6144
*b^2*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^(-6*d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x +
c)/d - (672*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 32*e^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) - 1/192*a*b
*((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)

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Fricas [A]  time = 1.97267, size = 589, normalized size = 4.03 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \,{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} + 80 \,{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 12 \,{\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} d x + 3 \,{\left (b^{2} \cosh \left (d x + c\right )^{7} + 8 \,{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{5} + 4 \,{\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 8 \,{\left (8 \, a^{2} - 15 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/384*(3*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(7*b^2*cosh(d*x + c)^3 + 8*(a*b - b^2)*cosh(d*x + c))*sinh(d*x
+ c)^5 + (21*b^2*cosh(d*x + c)^5 + 80*(a*b - b^2)*cosh(d*x + c)^3 + 12*(4*a^2 - 12*a*b + 7*b^2)*cosh(d*x + c))
*sinh(d*x + c)^3 + 3*(48*a^2 - 80*a*b + 35*b^2)*d*x + 3*(b^2*cosh(d*x + c)^7 + 8*(a*b - b^2)*cosh(d*x + c)^5 +
4*(4*a^2 - 12*a*b + 7*b^2)*cosh(d*x + c)^3 - 8*(8*a^2 - 15*a*b + 7*b^2)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 15.3862, size = 490, normalized size = 3.36 \begin{align*} \begin{cases} \frac{3 a^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 a^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 a^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 a^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac{15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac{15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac{11 a b \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 a b \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac{35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac{93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} - \frac{511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac{385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac{35 b^{2} \sinh{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \sinh ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**4*(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Piecewise((3*a**2*x*sinh(c + d*x)**4/8 - 3*a**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a**2*x*cosh(c + d*x)
**4/8 + 5*a**2*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*a**2*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 5*a*b*x*si
nh(c + d*x)**6/8 - 15*a*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/8 + 15*a*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/8
- 5*a*b*x*cosh(c + d*x)**6/8 + 11*a*b*sinh(c + d*x)**5*cosh(c + d*x)/(8*d) - 5*a*b*sinh(c + d*x)**3*cosh(c +
d*x)**3/(3*d) + 5*a*b*sinh(c + d*x)*cosh(c + d*x)**5/(8*d) + 35*b**2*x*sinh(c + d*x)**8/128 - 35*b**2*x*sinh(c
+ d*x)**6*cosh(c + d*x)**2/32 + 105*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 35*b**2*x*sinh(c + d*x)**2*
cosh(c + d*x)**6/32 + 35*b**2*x*cosh(c + d*x)**8/128 + 93*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*b*
*2*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) + 385*b**2*sinh(c + d*x)**3*cosh(c + d*x)**5/(384*d) - 35*b**2*si
nh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**2*sinh(c)**4, True))

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Giac [B]  time = 1.3632, size = 429, normalized size = 2.94 \begin{align*} \frac{3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 32 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 32 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 768 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 1440 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 672 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \,{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )}{\left (d x + c\right )} -{\left (2400 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 4000 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1750 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 768 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1440 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 672 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 32 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{6144 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/6144*(3*b^2*e^(8*d*x + 8*c) + 32*a*b*e^(6*d*x + 6*c) - 32*b^2*e^(6*d*x + 6*c) + 96*a^2*e^(4*d*x + 4*c) - 288
*a*b*e^(4*d*x + 4*c) + 168*b^2*e^(4*d*x + 4*c) - 768*a^2*e^(2*d*x + 2*c) + 1440*a*b*e^(2*d*x + 2*c) - 672*b^2*
e^(2*d*x + 2*c) + 48*(48*a^2 - 80*a*b + 35*b^2)*(d*x + c) - (2400*a^2*e^(8*d*x + 8*c) - 4000*a*b*e^(8*d*x + 8*
c) + 1750*b^2*e^(8*d*x + 8*c) - 768*a^2*e^(6*d*x + 6*c) + 1440*a*b*e^(6*d*x + 6*c) - 672*b^2*e^(6*d*x + 6*c) +
96*a^2*e^(4*d*x + 4*c) - 288*a*b*e^(4*d*x + 4*c) + 168*b^2*e^(4*d*x + 4*c) + 32*a*b*e^(2*d*x + 2*c) - 32*b^2*
e^(2*d*x + 2*c) + 3*b^2)*e^(-8*d*x - 8*c))/d