3.141 \(\int \sinh ^4(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=106 \[ \frac {a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 a x}{8}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {b \cosh (c+d x)}{d} \]

[Out]

3/8*a*x-b*cosh(d*x+c)/d+b*cosh(d*x+c)^3/d-3/5*b*cosh(d*x+c)^5/d+1/7*b*cosh(d*x+c)^7/d-3/8*a*cosh(d*x+c)*sinh(d
*x+c)/d+1/4*a*cosh(d*x+c)*sinh(d*x+c)^3/d

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Rubi [A]  time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3220, 2635, 8, 2633} \[ \frac {a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 a x}{8}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {b \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x)/8 - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/d - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7
*d) - (3*a*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

\begin {align*} \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=\int \left (a \sinh ^4(c+d x)+b \sinh ^7(c+d x)\right ) \, dx\\ &=a \int \sinh ^4(c+d x) \, dx+b \int \sinh ^7(c+d x) \, dx\\ &=\frac {a \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{4} (3 a) \int \sinh ^2(c+d x) \, dx-\frac {b \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 a \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {1}{8} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{8}-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 a \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 81, normalized size = 0.76 \[ \frac {-560 a \sinh (2 (c+d x))+70 a \sinh (4 (c+d x))+840 a c+840 a d x-1225 b \cosh (c+d x)+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x))}{2240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(840*a*c + 840*a*d*x - 1225*b*Cosh[c + d*x] + 245*b*Cosh[3*(c + d*x)] - 49*b*Cosh[5*(c + d*x)] + 5*b*Cosh[7*(c
+ d*x)] - 560*a*Sinh[2*(c + d*x)] + 70*a*Sinh[4*(c + d*x)])/(2240*d)

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fricas [A]  time = 1.54, size = 188, normalized size = 1.77 \[ \frac {5 \, b \cosh \left (d x + c\right )^{7} + 35 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 49 \, b \cosh \left (d x + c\right )^{5} + 280 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 35 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 7 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 245 \, b \cosh \left (d x + c\right )^{3} + 840 \, a d x + 35 \, {\left (3 \, b \cosh \left (d x + c\right )^{5} - 14 \, b \cosh \left (d x + c\right )^{3} + 21 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1225 \, b \cosh \left (d x + c\right ) + 280 \, {\left (a \cosh \left (d x + c\right )^{3} - 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2240*(5*b*cosh(d*x + c)^7 + 35*b*cosh(d*x + c)*sinh(d*x + c)^6 - 49*b*cosh(d*x + c)^5 + 280*a*cosh(d*x + c)*
sinh(d*x + c)^3 + 35*(5*b*cosh(d*x + c)^3 - 7*b*cosh(d*x + c))*sinh(d*x + c)^4 + 245*b*cosh(d*x + c)^3 + 840*a
*d*x + 35*(3*b*cosh(d*x + c)^5 - 14*b*cosh(d*x + c)^3 + 21*b*cosh(d*x + c))*sinh(d*x + c)^2 - 1225*b*cosh(d*x
+ c) + 280*(a*cosh(d*x + c)^3 - 4*a*cosh(d*x + c))*sinh(d*x + c))/d

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giac [A]  time = 0.16, size = 182, normalized size = 1.72 \[ \frac {3}{8} \, a x + \frac {b e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {7 \, b e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {a e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {7 \, b e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} - \frac {a e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac {35 \, b e^{\left (d x + c\right )}}{128 \, d} - \frac {35 \, b e^{\left (-d x - c\right )}}{128 \, d} + \frac {a e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} + \frac {7 \, b e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} - \frac {a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} - \frac {7 \, b e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {b e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*a*x + 1/896*b*e^(7*d*x + 7*c)/d - 7/640*b*e^(5*d*x + 5*c)/d + 1/64*a*e^(4*d*x + 4*c)/d + 7/128*b*e^(3*d*x
+ 3*c)/d - 1/8*a*e^(2*d*x + 2*c)/d - 35/128*b*e^(d*x + c)/d - 35/128*b*e^(-d*x - c)/d + 1/8*a*e^(-2*d*x - 2*c)
/d + 7/128*b*e^(-3*d*x - 3*c)/d - 1/64*a*e^(-4*d*x - 4*c)/d - 7/640*b*e^(-5*d*x - 5*c)/d + 1/896*b*e^(-7*d*x -
7*c)/d

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maple [A]  time = 0.04, size = 82, normalized size = 0.77 \[ \frac {b \left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{35}\right ) \cosh \left (d x +c \right )+a \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),x)

[Out]

1/d*(b*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+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 = 0.32, size = 164, normalized size = 1.55 \[ \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{4480} \, b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),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) - 1/4480*b
*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x -
c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d)

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mupad [B]  time = 0.26, size = 85, normalized size = 0.80 \[ \frac {280\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3-280\,b\,\mathrm {cosh}\left (c+d\,x\right )-168\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5+40\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7-175\,a\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )+105\,a\,d\,x+70\,a\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )}{280\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^4*(a + b*sinh(c + d*x)^3),x)

[Out]

(280*b*cosh(c + d*x)^3 - 280*b*cosh(c + d*x) - 168*b*cosh(c + d*x)^5 + 40*b*cosh(c + d*x)^7 - 175*a*cosh(c + d
*x)*sinh(c + d*x) + 105*a*d*x + 70*a*cosh(c + d*x)^3*sinh(c + d*x))/(280*d)

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sympy [A]  time = 5.12, size = 192, normalized size = 1.81 \[ \begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\relax (c )}\right ) \sinh ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**4*(a+b*sinh(d*x+c)**3),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) + b*sinh(c + d*x)**6*cosh(c
+ d*x)/d - 2*b*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 8*b*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 16*b*cosh(
c + d*x)**7/(35*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)*sinh(c)**4, True))

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