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

Optimal. Leaf size=28 \[ \frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0216874, antiderivative size = 28, 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 = {3190} \[ \frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

Mathematica [A]  time = 0.0117645, size = 39, normalized size = 1.39 \[ \frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0335, size = 35, normalized size = 1.25 \begin{align*} \frac{b \sinh \left (d x + c\right )^{3}}{3 \, d} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.4489, size = 103, normalized size = 3.68 \begin{align*} \frac{b \sinh \left (d x + c\right )^{3} + 3 \,{\left (b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.487066, size = 36, normalized size = 1.29 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b \sinh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*(a+b*sinh(d*x+c)**2),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.13263, size = 97, normalized size = 3.46 \begin{align*} \frac{b e^{\left (3 \, d x + 3 \, c\right )} + 12 \, a e^{\left (d x + c\right )} - 3 \, b e^{\left (d x + c\right )} -{\left (12 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, 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(cosh(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) - 3*b*e^(d*x + c) - (12*a*e^(2*d*x + 2*c) - 3*b*e^(2*d*x + 2*c) + b
)*e^(-3*d*x - 3*c))/d