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

Optimal. Leaf size=92 \[ \frac{2 b \left (4 a^2-b^2\right ) \cosh (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2-3 b^2\right )+\frac{5 a b^2 \sinh (c+d x) \cosh (c+d x)}{6 d}+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d} \]

[Out]

(a*(2*a^2 - 3*b^2)*x)/2 + (2*b*(4*a^2 - b^2)*Cosh[c + d*x])/(3*d) + (5*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(6*d
) + (b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(3*d)

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Rubi [A]  time = 0.0728995, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2656, 2734} \[ \frac{2 b \left (4 a^2-b^2\right ) \cosh (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2-3 b^2\right )+\frac{5 a b^2 \sinh (c+d x) \cosh (c+d x)}{6 d}+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x])^3,x]

[Out]

(a*(2*a^2 - 3*b^2)*x)/2 + (2*b*(4*a^2 - b^2)*Cosh[c + d*x])/(3*d) + (5*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(6*d
) + (b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(3*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 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))^3 \, dx &=\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d}+\frac{1}{3} \int (a+b \sinh (c+d x)) \left (3 a^2-2 b^2+5 a b \sinh (c+d x)\right ) \, dx\\ &=\frac{1}{2} a \left (2 a^2-3 b^2\right ) x+\frac{2 b \left (4 a^2-b^2\right ) \cosh (c+d x)}{3 d}+\frac{5 a b^2 \cosh (c+d x) \sinh (c+d x)}{6 d}+\frac{b \cosh (c+d x) (a+b \sinh (c+d x))^2}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.174541, size = 71, normalized size = 0.77 \[ \frac{6 a \left (2 a^2-3 b^2\right ) (c+d x)-9 b \left (b^2-4 a^2\right ) \cosh (c+d x)+9 a b^2 \sinh (2 (c+d x))+b^3 \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x])^3,x]

[Out]

(6*a*(2*a^2 - 3*b^2)*(c + d*x) - 9*b*(-4*a^2 + b^2)*Cosh[c + d*x] + b^3*Cosh[3*(c + d*x)] + 9*a*b^2*Sinh[2*(c
+ d*x)])/(12*d)

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Maple [A]  time = 0.017, size = 77, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +3\,{a}^{2}b\cosh \left ( dx+c \right ) +{a}^{3} \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))^3,x)

[Out]

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

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Maxima [A]  time = 1.02683, size = 155, normalized size = 1.68 \begin{align*} -\frac{3}{8} \, a 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^{3} x + \frac{1}{24} \, 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{3 \, a^{2} 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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94243, size = 225, normalized size = 2.45 \begin{align*} \frac{b^{3} \cosh \left (d x + c\right )^{3} + 3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 18 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 6 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} d x + 9 \,{\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b^3*cosh(d*x + c)^3 + 3*b^3*cosh(d*x + c)*sinh(d*x + c)^2 + 18*a*b^2*cosh(d*x + c)*sinh(d*x + c) + 6*(2*
a^3 - 3*a*b^2)*d*x + 9*(4*a^2*b - b^3)*cosh(d*x + c))/d

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Sympy [A]  time = 0.753248, size = 128, normalized size = 1.39 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b \cosh{\left (c + d x \right )}}{d} + \frac{3 a b^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{3 a b^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} + \frac{b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c))**3,x)

[Out]

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

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Giac [A]  time = 1.32152, size = 173, normalized size = 1.88 \begin{align*} \frac{b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 36 \, a^{2} b e^{\left (d x + c\right )} - 9 \, b^{3} e^{\left (d x + c\right )} + 12 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (9 \, a b^{2} e^{\left (d x + c\right )} - b^{3} - 9 \,{\left (4 \, a^{2} b - b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^3*e^(3*d*x + 3*c) + 9*a*b^2*e^(2*d*x + 2*c) + 36*a^2*b*e^(d*x + c) - 9*b^3*e^(d*x + c) + 12*(2*a^3 - 3
*a*b^2)*(d*x + c) - (9*a*b^2*e^(d*x + c) - b^3 - 9*(4*a^2*b - b^3)*e^(2*d*x + 2*c))*e^(-3*d*x - 3*c))/d