3.284 \(\int \cosh ^3(c+d x) (a+b \sinh ^2(c+d x)) \, dx\)

Optimal. Leaf size=46 \[ \frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^5(c+d x)}{5 d} \]

[Out]

(a*Sinh[c + d*x])/d + ((a + b)*Sinh[c + d*x]^3)/(3*d) + (b*Sinh[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.0416845, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3190, 373} \[ \frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]

[Out]

(a*Sinh[c + d*x])/d + ((a + b)*Sinh[c + d*x]^3)/(3*d) + (b*Sinh[c + d*x]^5)/(5*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]

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.139109, size = 48, normalized size = 1.04 \[ \frac{\sinh (c+d x) (4 (5 a+2 b) \cosh (2 (c+d x))+100 a+3 b \cosh (4 (c+d x))-11 b)}{120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]

[Out]

((100*a - 11*b + 4*(5*a + 2*b)*Cosh[2*(c + d*x)] + 3*b*Cosh[4*(c + d*x)])*Sinh[c + d*x])/(120*d)

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Maple [A]  time = 0.053, size = 65, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{\sinh \left ( dx+c \right ) }{5} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +a \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x)

[Out]

1/d*(b*(1/5*sinh(d*x+c)*cosh(d*x+c)^4-1/5*(2/3+1/3*cosh(d*x+c)^2)*sinh(d*x+c))+a*(2/3+1/3*cosh(d*x+c)^2)*sinh(
d*x+c))

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Maxima [B]  time = 1.01288, size = 184, normalized size = 4. \begin{align*} \frac{1}{480} \, b{\left (\frac{{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac{30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac{1}{24} \, a{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*b*((5*e^(-2*d*x - 2*c) - 30*e^(-4*d*x - 4*c) + 3)*e^(5*d*x + 5*c)/d + (30*e^(-d*x - c) - 5*e^(-3*d*x - 3
*c) - 3*e^(-5*d*x - 5*c))/d) + 1/24*a*(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)

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Fricas [A]  time = 1.48221, size = 220, normalized size = 4.78 \begin{align*} \frac{3 \, b \sinh \left (d x + c\right )^{5} + 5 \,{\left (6 \, b \cosh \left (d x + c\right )^{2} + 4 \, a + b\right )} \sinh \left (d x + c\right )^{3} + 15 \,{\left (b \cosh \left (d x + c\right )^{4} +{\left (4 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 12 \, a - 2 \, b\right )} \sinh \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(3*b*sinh(d*x + c)^5 + 5*(6*b*cosh(d*x + c)^2 + 4*a + b)*sinh(d*x + c)^3 + 15*(b*cosh(d*x + c)^4 + (4*a
+ b)*cosh(d*x + c)^2 + 12*a - 2*b)*sinh(d*x + c))/d

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Sympy [A]  time = 2.04711, size = 85, normalized size = 1.85 \begin{align*} \begin{cases} - \frac{2 a \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac{a \sinh{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac{2 b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac{b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{3}{\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)**3*(a+b*sinh(d*x+c)**2),x)

[Out]

Piecewise((-2*a*sinh(c + d*x)**3/(3*d) + a*sinh(c + d*x)*cosh(c + d*x)**2/d - 2*b*sinh(c + d*x)**5/(15*d) + b*
sinh(c + d*x)**3*cosh(c + d*x)**2/(3*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*cosh(c)**3, True))

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Giac [B]  time = 1.17355, size = 166, normalized size = 3.61 \begin{align*} \frac{3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 20 \, a e^{\left (3 \, d x + 3 \, c\right )} + 5 \, b e^{\left (3 \, d x + 3 \, c\right )} + 180 \, a e^{\left (d x + c\right )} - 30 \, b e^{\left (d x + c\right )} -{\left (180 \, a e^{\left (4 \, d x + 4 \, c\right )} - 30 \, b e^{\left (4 \, d x + 4 \, c\right )} + 20 \, a e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/480*(3*b*e^(5*d*x + 5*c) + 20*a*e^(3*d*x + 3*c) + 5*b*e^(3*d*x + 3*c) + 180*a*e^(d*x + c) - 30*b*e^(d*x + c)
- (180*a*e^(4*d*x + 4*c) - 30*b*e^(4*d*x + 4*c) + 20*a*e^(2*d*x + 2*c) + 5*b*e^(2*d*x + 2*c) + 3*b)*e^(-5*d*x
- 5*c))/d