3.4 \(\int \sinh (c+d x) (a+b \sinh ^2(c+d x)) \, dx\)

Optimal. Leaf size=32 \[ \frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d} \]

[Out]

((a - b)*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0289156, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3013} \[ \frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2),x]

[Out]

((a - b)*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)

Rule 3013

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Dist[f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rubi steps

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

Mathematica [A]  time = 0.0240816, size = 53, normalized size = 1.66 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +a\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)*(a+b*sinh(d*x+c)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.0147, size = 90, normalized size = 2.81 \begin{align*} \frac{1}{24} \, b{\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{a \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/24*b*(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) + a*cosh(d*x + c)/d

________________________________________________________________________________________

Fricas [A]  time = 1.87512, size = 127, normalized size = 3.97 \begin{align*} \frac{b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \,{\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.768842, size = 56, normalized size = 1.75 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{d} + \frac{b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \sinh{\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)*(a+b*sinh(d*x+c)**2),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.26747, size = 96, normalized size = 3. \begin{align*} \frac{b e^{\left (3 \, d x + 3 \, c\right )} + 12 \, a e^{\left (d x + c\right )} - 9 \, b e^{\left (d x + c\right )} +{\left (12 \, a e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\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(sinh(d*x+c)*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

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