3.2.84 \(\int \sinh ^4(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [184]

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

[Out]

1/128*(48*a+35*b)*x-1/128*(80*a+93*b)*cosh(d*x+c)*sinh(d*x+c)/d+1/192*(48*a+163*b)*cosh(d*x+c)^3*sinh(d*x+c)/d
-25/48*b*cosh(d*x+c)^5*sinh(d*x+c)/d+1/8*b*cosh(d*x+c)^7*sinh(d*x+c)/d

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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {3296, 1271, 1828, 1171, 393, 212} \begin {gather*} \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} \end {gather*}

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 212

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

Rule 393

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 1171

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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1271

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/(d + e*x^2))*(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))], 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 1828

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 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 \sinh ^4(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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) \text {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*}

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Mathematica [A]
time = 0.11, size = 82, normalized size = 0.74 \begin {gather*} \frac {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} \end {gather*}

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 = 1.13, size = 82, normalized size = 0.74

method	result	size
default	\(\frac {\left (-\frac {7 b}{16}-\frac {a}{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {\left (\frac {7 b}{32}+\frac {a}{8}\right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {3 a x}{8}+\frac {35 b x}{128}-\frac {b \sinh \left (6 d x +6 c \right )}{96 d}+\frac {b \sinh \left (8 d x +8 c \right )}{1024 d}\)	\(82\)
risch	\(\frac {35 b x}{128}+\frac {3 a x}{8}+\frac {b \,{\mathrm e}^{8 d x +8 c}}{2048 d}-\frac {b \,{\mathrm e}^{6 d x +6 c}}{192 d}+\frac {7 \,{\mathrm e}^{4 d x +4 c} b}{256 d}+\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}-\frac {7 b \,{\mathrm e}^{2 d x +2 c}}{64 d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}+\frac {7 \,{\mathrm e}^{-2 d x -2 c} b}{64 d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {7 \,{\mathrm e}^{-4 d x -4 c} b}{256 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}+\frac {b \,{\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {b \,{\mathrm e}^{-8 d x -8 c}}{2048 d}\)	\(190\)

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,method=_RETURNVERBOSE)

[Out]

1/2*(-7/16*b-1/2*a)*sinh(2*d*x+2*c)/d+1/4*(7/32*b+1/8*a)*sinh(4*d*x+4*c)/d+3/8*a*x+35/128*b*x-1/96*b*sinh(6*d*
x+6*c)/d+1/1024*b*sinh(8*d*x+8*c)/d

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Maxima [A]
time = 0.27, size = 175, normalized size = 1.58 \begin {gather*} \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 {gather*}

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 = 0.44, size = 174, normalized size = 1.57 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (104) = 208\).
time = 1.22, size = 306, normalized size = 2.76 \begin {gather*} \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 {gather*}

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 = 0.44, size = 155, normalized size = 1.40 \begin {gather*} \frac {1}{128} \, {\left (48 \, a + 35 \, b\right )} x + \frac {b e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {{\left (4 \, a + 7 \, b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} - \frac {{\left (8 \, a + 7 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} + \frac {{\left (8 \, a + 7 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {{\left (4 \, a + 7 \, b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} + \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {b e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} \end {gather*}

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]

1/128*(48*a + 35*b)*x + 1/2048*b*e^(8*d*x + 8*c)/d - 1/192*b*e^(6*d*x + 6*c)/d + 1/256*(4*a + 7*b)*e^(4*d*x +
4*c)/d - 1/64*(8*a + 7*b)*e^(2*d*x + 2*c)/d + 1/64*(8*a + 7*b)*e^(-2*d*x - 2*c)/d - 1/256*(4*a + 7*b)*e^(-4*d*
x - 4*c)/d + 1/192*b*e^(-6*d*x - 6*c)/d - 1/2048*b*e^(-8*d*x - 8*c)/d

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Mupad [B]
time = 0.92, size = 88, normalized size = 0.79 \begin {gather*} \frac {12\,a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-96\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-84\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+21\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-4\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {3\,b\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+144\,a\,d\,x+105\,b\,d\,x}{384\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*a*sinh(4*c + 4*d*x) - 96*a*sinh(2*c + 2*d*x) - 84*b*sinh(2*c + 2*d*x) + 21*b*sinh(4*c + 4*d*x) - 4*b*sinh(
6*c + 6*d*x) + (3*b*sinh(8*c + 8*d*x))/8 + 144*a*d*x + 105*b*d*x)/(384*d)

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