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

Optimal. Leaf size=61 \[ \frac{(4 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (4 a-b)+\frac{b \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]

[Out]

((4*a - b)*x)/8 + ((4*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)

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Rubi [A]  time = 0.0488858, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3191, 385, 199, 206} \[ \frac{(4 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (4 a-b)+\frac{b \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a - b)*x)/8 + ((4*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*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 ^2(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 )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{(4 a-b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{(4 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{(4 a-b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{1}{8} (4 a-b) x+\frac{(4 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0695163, size = 43, normalized size = 0.7 \[ \frac{8 a \sinh (2 (c+d x))+16 a c+16 a d x+b \sinh (4 (c+d x))-4 b d x}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*c + 16*a*d*x - 4*b*d*x + 8*a*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)])/(32*d)

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Maple [A]  time = 0.026, size = 70, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{8}}-{\frac{dx}{8}}-{\frac{c}{8}} \right ) +a \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00693, size = 103, normalized size = 1.69 \begin{align*} \frac{1}{8} \, a{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{64} \, b{\left (\frac{8 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*(4*x + e^(2*d*x + 2*c)/d - e^(-2*d*x - 2*c)/d) - 1/64*b*(8*(d*x + c)/d - e^(4*d*x + 4*c)/d + e^(-4*d*x -
4*c)/d)

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Fricas [A]  time = 1.45926, size = 153, normalized size = 2.51 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (4 \, a - b\right )} d x +{\left (b \cosh \left (d x + c\right )^{3} + 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.14033, size = 150, normalized size = 2.46 \begin{align*} \begin{cases} - \frac{a x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{a \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} - \frac{b x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac{b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac{b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{2}{\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)**2*(a+b*sinh(d*x+c)**2),x)

[Out]

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

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Giac [A]  time = 1.12573, size = 124, normalized size = 2.03 \begin{align*} \frac{8 \,{\left (d x + c\right )}{\left (4 \, a - b\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} -{\left (24 \, a e^{\left (4 \, d x + 4 \, c\right )} - 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(8*(d*x + c)*(4*a - b) + b*e^(4*d*x + 4*c) + 8*a*e^(2*d*x + 2*c) - (24*a*e^(4*d*x + 4*c) - 6*b*e^(4*d*x +
4*c) + 8*a*e^(2*d*x + 2*c) + b)*e^(-4*d*x - 4*c))/d