3.283 \(\int \cosh ^4(c+d x) (a+b \sinh ^2(c+d x)) \, dx\)

Optimal. Leaf size=89 \[ \frac{(6 a-b) \sinh (c+d x) \cosh ^3(c+d x)}{24 d}+\frac{(6 a-b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac{1}{16} x (6 a-b)+\frac{b \sinh (c+d x) \cosh ^5(c+d x)}{6 d} \]

[Out]

((6*a - b)*x)/16 + ((6*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - b)*Cosh[c + d*x]^3*Sinh[c + d*x])/
(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)

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Rubi [A]  time = 0.0571499, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3191, 385, 199, 206} \[ \frac{(6 a-b) \sinh (c+d x) \cosh ^3(c+d x)}{24 d}+\frac{(6 a-b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac{1}{16} x (6 a-b)+\frac{b \sinh (c+d x) \cosh ^5(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]

[Out]

((6*a - b)*x)/16 + ((6*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - b)*Cosh[c + d*x]^3*Sinh[c + d*x])/
(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)

Rule 3191

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-(a-b) x^2}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac{(6 a-b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=\frac{(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac{(6 a-b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac{(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac{(6 a-b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=\frac{1}{16} (6 a-b) x+\frac{(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac{(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.168358, size = 63, normalized size = 0.71 \[ \frac{(48 a-3 b) \sinh (2 (c+d x))+3 (2 a+b) \sinh (4 (c+d x))+72 a c+72 a d x+b \sinh (6 (c+d x))-12 b d x}{192 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]

[Out]

(72*a*c + 72*a*d*x - 12*b*d*x + (48*a - 3*b)*Sinh[2*(c + d*x)] + 3*(2*a + b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c +
d*x)])/(192*d)

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Maple [A]  time = 0.059, size = 95, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{6}}-{\frac{\sinh \left ( dx+c \right ) }{6} \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) }-{\frac{dx}{16}}-{\frac{c}{16}} \right ) +a \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x)

[Out]

1/d*(b*(1/6*sinh(d*x+c)*cosh(d*x+c)^5-1/6*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)-1/16*d*x-1/16*c)+a*(
(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.01327, size = 205, normalized size = 2.3 \begin{align*} \frac{1}{64} \, a{\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{1}{384} \, b{\left (\frac{{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac{24 \,{\left (d x + c\right )}}{d} + \frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/64*a*(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) + 1/384*b*
((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + 1)*e^(6*d*x + 6*c)/d - 24*(d*x + c)/d + (3*e^(-2*d*x - 2*c) - 3*e^
(-4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)

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Fricas [A]  time = 1.48793, size = 306, normalized size = 3.44 \begin{align*} \frac{3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \,{\left (5 \, b \cosh \left (d x + c\right )^{3} + 3 \,{\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \,{\left (6 \, a - b\right )} d x + 3 \,{\left (b \cosh \left (d x + c\right )^{5} + 2 \,{\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} +{\left (16 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 + 3*(2*a + b)*cosh(d*x + c))*sinh(d*x + c)^3
+ 6*(6*a - b)*d*x + 3*(b*cosh(d*x + c)^5 + 2*(2*a + b)*cosh(d*x + c)^3 + (16*a - b)*cosh(d*x + c))*sinh(d*x +
c))/d

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Sympy [A]  time = 4.28559, size = 250, normalized size = 2.81 \begin{align*} \begin{cases} \frac{3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 a x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{5 a \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac{3 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac{3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac{b x \cosh ^{6}{\left (c + d x \right )}}{16} - \frac{b \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{16 d} + \frac{b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac{b \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{4}{\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)**4*(a+b*sinh(d*x+c)**2),x)

[Out]

Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a*x*cosh(c + d*x)**4/8 - 3
*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + b*x*sinh(c + d*x)**6/16 -
3*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 3*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - b*x*cosh(c + d*x)**
6/16 - b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) + b*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + b*sinh(c + d*x)*c
osh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*cosh(c)**4, True))

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Giac [B]  time = 1.20001, size = 221, normalized size = 2.48 \begin{align*} \frac{24 \,{\left (d x + c\right )}{\left (6 \, a - b\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} -{\left (132 \, a e^{\left (6 \, d x + 6 \, c\right )} - 22 \, b e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/384*(24*(d*x + c)*(6*a - b) + b*e^(6*d*x + 6*c) + 6*a*e^(4*d*x + 4*c) + 3*b*e^(4*d*x + 4*c) + 48*a*e^(2*d*x
+ 2*c) - 3*b*e^(2*d*x + 2*c) - (132*a*e^(6*d*x + 6*c) - 22*b*e^(6*d*x + 6*c) + 48*a*e^(4*d*x + 4*c) - 3*b*e^(4
*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + b)*e^(-6*d*x - 6*c))/d