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

Optimal. Leaf size=137 \[ \frac{a b \left (19 a^2-16 b^2\right ) \cosh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2-9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{24 d}+\frac{1}{8} x \left (-24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac{7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{12 d} \]

[Out]

((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 - 16*b^2)*Cosh[c + d*x])/(6*d) + (b^2*(26*a^2 - 9*b^2)*Cosh[
c + d*x]*Sinh[c + d*x])/(24*d) + (7*a*b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(12*d) + (b*Cosh[c + d*x]*(a +
b*Sinh[c + d*x])^3)/(4*d)

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Rubi [A]  time = 0.155736, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2656, 2753, 2734} \[ \frac{a b \left (19 a^2-16 b^2\right ) \cosh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2-9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{24 d}+\frac{1}{8} x \left (-24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac{7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{12 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x])^4,x]

[Out]

((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 - 16*b^2)*Cosh[c + d*x])/(6*d) + (b^2*(26*a^2 - 9*b^2)*Cosh[
c + d*x]*Sinh[c + d*x])/(24*d) + (7*a*b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(12*d) + (b*Cosh[c + d*x]*(a +
b*Sinh[c + d*x])^3)/(4*d)

Rule 2656

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[1/n, Int[(a + b*Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*
x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int (a+b \sinh (c+d x))^4 \, dx &=\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac{1}{4} \int (a+b \sinh (c+d x))^2 \left (4 a^2-3 b^2+7 a b \sinh (c+d x)\right ) \, dx\\ &=\frac{7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{12 d}+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}+\frac{1}{12} \int (a+b \sinh (c+d x)) \left (a \left (12 a^2-23 b^2\right )+b \left (26 a^2-9 b^2\right ) \sinh (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a^4-24 a^2 b^2+3 b^4\right ) x+\frac{a b \left (19 a^2-16 b^2\right ) \cosh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2-9 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{24 d}+\frac{7 a b \cosh (c+d x) (a+b \sinh (c+d x))^2}{12 d}+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^3}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.393612, size = 108, normalized size = 0.79 \[ \frac{3 \left (4 \left (-24 a^2 b^2+8 a^4+3 b^4\right ) (c+d x)+8 \left (6 a^2 b^2-b^4\right ) \sinh (2 (c+d x))+b^4 \sinh (4 (c+d x))\right )+96 a b \left (4 a^2-3 b^2\right ) \cosh (c+d x)+32 a b^3 \cosh (3 (c+d x))}{96 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x])^4,x]

[Out]

(96*a*b*(4*a^2 - 3*b^2)*Cosh[c + d*x] + 32*a*b^3*Cosh[3*(c + d*x)] + 3*(4*(8*a^4 - 24*a^2*b^2 + 3*b^4)*(c + d*
x) + 8*(6*a^2*b^2 - b^4)*Sinh[2*(c + d*x)] + b^4*Sinh[4*(c + d*x)]))/(96*d)

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Maple [A]  time = 0.016, size = 119, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{4} \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 ) +4\,a{b}^{3} \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +6\,{a}^{2}{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +4\,{a}^{3}b\cosh \left ( dx+c \right ) +{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^4*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+4*a*b^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh
(d*x+c)+6*a^2*b^2*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+4*a^3*b*cosh(d*x+c)+a^4*(d*x+c))

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Maxima [A]  time = 1.10724, size = 246, normalized size = 1.8 \begin{align*} \frac{1}{64} \, b^{4}{\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{3}{4} \, a^{2} b^{2}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{4} x + \frac{1}{6} \, a b^{3}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{4 \, a^{3} b \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*b^4*(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) - 3/4*a^
2*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + a^4*x + 1/6*a*b^3*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d
- 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 4*a^3*b*cosh(d*x + c)/d

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Fricas [A]  time = 2.05994, size = 358, normalized size = 2.61 \begin{align*} \frac{3 \, b^{4} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 8 \, a b^{3} \cosh \left (d x + c\right )^{3} + 24 \, a b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \,{\left (8 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 24 \,{\left (4 \, a^{3} b - 3 \, a b^{3}\right )} \cosh \left (d x + c\right ) + 3 \,{\left (b^{4} \cosh \left (d x + c\right )^{3} + 4 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^4*cosh(d*x + c)*sinh(d*x + c)^3 + 8*a*b^3*cosh(d*x + c)^3 + 24*a*b^3*cosh(d*x + c)*sinh(d*x + c)^2 +
3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*d*x + 24*(4*a^3*b - 3*a*b^3)*cosh(d*x + c) + 3*(b^4*cosh(d*x + c)^3 + 4*(6*a^2
*b^2 - b^4)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 1.48116, size = 240, normalized size = 1.75 \begin{align*} \begin{cases} a^{4} x + \frac{4 a^{3} b \cosh{\left (c + d x \right )}}{d} + 3 a^{2} b^{2} x \sinh ^{2}{\left (c + d x \right )} - 3 a^{2} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac{3 a^{2} b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} + \frac{4 a b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 a b^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{3 b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 b^{4} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*x + 4*a**3*b*cosh(c + d*x)/d + 3*a**2*b**2*x*sinh(c + d*x)**2 - 3*a**2*b**2*x*cosh(c + d*x)**2
+ 3*a**2*b**2*sinh(c + d*x)*cosh(c + d*x)/d + 4*a*b**3*sinh(c + d*x)**2*cosh(c + d*x)/d - 8*a*b**3*cosh(c + d
*x)**3/(3*d) + 3*b**4*x*sinh(c + d*x)**4/8 - 3*b**4*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*b**4*x*cosh(c +
d*x)**4/8 + 5*b**4*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*b**4*sinh(c + d*x)*cosh(c + d*x)**3/(8*d), Ne(d, 0
)), (x*(a + b*sinh(c))**4, True))

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Giac [A]  time = 1.18111, size = 261, normalized size = 1.91 \begin{align*} \frac{3 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 144 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 384 \, a^{3} b e^{\left (d x + c\right )} - 288 \, a b^{3} e^{\left (d x + c\right )} + 24 \,{\left (8 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )}{\left (d x + c\right )} +{\left (32 \, a b^{3} e^{\left (d x + c\right )} - 3 \, b^{4} + 96 \,{\left (4 \, a^{3} b - 3 \, a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} - 24 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b^4*e^(4*d*x + 4*c) + 32*a*b^3*e^(3*d*x + 3*c) + 144*a^2*b^2*e^(2*d*x + 2*c) - 24*b^4*e^(2*d*x + 2*c)
+ 384*a^3*b*e^(d*x + c) - 288*a*b^3*e^(d*x + c) + 24*(8*a^4 - 24*a^2*b^2 + 3*b^4)*(d*x + c) + (32*a*b^3*e^(d*
x + c) - 3*b^4 + 96*(4*a^3*b - 3*a*b^3)*e^(3*d*x + 3*c) - 24*(6*a^2*b^2 - b^4)*e^(2*d*x + 2*c))*e^(-4*d*x - 4*
c))/d