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

Optimal. Leaf size=111 \[ \frac{(48 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{(80 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x (48 a+35 b)+\frac{b \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]

[Out]

((48*a + 35*b)*x)/128 - ((80*a + 93*b)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a + 163*b)*Cosh[c + d*x]^3*
Sinh[c + d*x])/(192*d) - (25*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x])/(8*d)

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Rubi [A]  time = 0.142796, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3217, 1257, 1814, 1157, 385, 206} \[ \frac{(48 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{(80 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x (48 a+35 b)+\frac{b \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*a + 35*b)*x)/128 - ((80*a + 93*b)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a + 163*b)*Cosh[c + d*x]^3*
Sinh[c + d*x])/(192*d) - (25*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x])/(8*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 1257

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[1/(2*e^(2*p +
m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4
)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(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] && IGtQ[m/2, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

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 ^4(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a-2 a x^2+(a+b) x^4\right )}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{b+8 b x^2-8 (a-b) x^4+8 (a+b) x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac{25 b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac{\operatorname{Subst}\left (\int \frac{19 b+96 b x^2+48 (a+b) x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac{(48 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{3 (16 a+29 b)+192 (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{192 d}\\ &=-\frac{(80 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{(48 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac{(48 a+35 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{1}{128} (48 a+35 b) x-\frac{(80 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{(48 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.141628, size = 82, normalized size = 0.74 \[ \frac{-96 (8 a+7 b) \sinh (2 (c+d x))+24 (4 a+7 b) \sinh (4 (c+d x))+1152 a c+1152 a d x-32 b \sinh (6 (c+d x))+3 b \sinh (8 (c+d x))+840 b c+840 b d x}{3072 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 98, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( b \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 ) +a \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)^4),x)

[Out]

1/d*(b*((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)+a*((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 = 1.14058, size = 236, normalized size = 2.13 \begin{align*} \frac{1}{64} \, a{\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{\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 )} \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)^4),x, algorithm="maxima")

[Out]

1/64*a*(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
*((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)

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Fricas [A]  time = 1.62985, size = 467, normalized size = 4.21 \begin{align*} \frac{3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b \cosh \left (d x + c\right )^{3} - 8 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b \cosh \left (d x + c\right )^{5} - 80 \, b \cosh \left (d x + c\right )^{3} + 12 \,{\left (4 \, a + 7 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (48 \, a + 35 \, b\right )} d x + 3 \,{\left (b \cosh \left (d x + c\right )^{7} - 8 \, b \cosh \left (d x + c\right )^{5} + 4 \,{\left (4 \, a + 7 \, b\right )} \cosh \left (d x + c\right )^{3} - 8 \,{\left (8 \, a + 7 \, b\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)^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 14.1661, size = 306, normalized size = 2.76 \begin{align*} \begin{cases} \frac{3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 a \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 a \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{35 b x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{35 b x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{105 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{35 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{35 b x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac{93 b \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} - \frac{511 b \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac{385 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac{35 b \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 ^{4}{\left (c \right )}\right ) \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)**4),x)

[Out]

Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a*x*cosh(c + d*x)**4/8 + 5
*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 35*b*x*sinh(c + d*x)**8/1
28 - 35*b*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 105*b*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 35*b*x*sinh(
c + d*x)**2*cosh(c + d*x)**6/32 + 35*b*x*cosh(c + d*x)**8/128 + 93*b*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) -
511*b*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) + 385*b*sinh(c + d*x)**3*cosh(c + d*x)**5/(384*d) - 35*b*sinh(
c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)*sinh(c)**4, True))

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Giac [A]  time = 1.20316, size = 258, normalized size = 2.32 \begin{align*} \frac{48 \,{\left (d x + c\right )}{\left (48 \, a + 35 \, b\right )} + 3 \, b e^{\left (8 \, d x + 8 \, c\right )} - 32 \, b e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b e^{\left (4 \, d x + 4 \, c\right )} - 768 \, a e^{\left (2 \, d x + 2 \, c\right )} - 672 \, b e^{\left (2 \, d x + 2 \, c\right )} -{\left (2400 \, a e^{\left (8 \, d x + 8 \, c\right )} + 1750 \, b e^{\left (8 \, d x + 8 \, c\right )} - 768 \, a e^{\left (6 \, d x + 6 \, c\right )} - 672 \, b e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b e^{\left (4 \, d x + 4 \, c\right )} - 32 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b\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)^4),x, algorithm="giac")

[Out]

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