3.60 \(\int x \cosh (a+b \sqrt{c+d x}) \, dx\)

Optimal. Leaf size=167 \[ \frac{12 \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{6 (c+d x) \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{2 c \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{12 \cosh \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2} \]

[Out]

(-12*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (2*c*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^2) - (6*(c + d*x)*Cosh[a +
b*Sqrt[c + d*x]])/(b^2*d^2) + (12*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*c*Sqrt[c + d*x]*Sinh
[a + b*Sqrt[c + d*x]])/(b*d^2) + (2*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^2)

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Rubi [A]  time = 0.188235, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5365, 5287, 3296, 2638} \[ \frac{12 \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{6 (c+d x) \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{2 c \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{12 \cosh \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*Sqrt[c + d*x]],x]

[Out]

(-12*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (2*c*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^2) - (6*(c + d*x)*Cosh[a +
b*Sqrt[c + d*x]])/(b^2*d^2) + (12*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*c*Sqrt[c + d*x]*Sinh
[a + b*Sqrt[c + d*x]])/(b*d^2) + (2*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^2)

Rule 5365

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rule 5287

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x \cosh \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (-c+x) \cosh \left (a+b \sqrt{x}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int x \left (-c+x^2\right ) \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-c x \cosh (a+b x)+x^3 \cosh (a+b x)\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^2}-\frac{(2 c) \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{6 \operatorname{Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^2}+\frac{(2 c) \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^2}\\ &=\frac{2 c \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{6 (c+d x) \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{12 \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^2}\\ &=\frac{2 c \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{6 (c+d x) \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{12 \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{12 \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^2}\\ &=-\frac{12 \cosh \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 c \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{6 (c+d x) \cosh \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{12 \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 c \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (c+d x)^{3/2} \sinh \left (a+b \sqrt{c+d x}\right )}{b d^2}\\ \end{align*}

Mathematica [A]  time = 0.189866, size = 72, normalized size = 0.43 \[ \frac{2 b \left (b^2 d x+6\right ) \sqrt{c+d x} \sinh \left (a+b \sqrt{c+d x}\right )-2 \left (b^2 (2 c+3 d x)+6\right ) \cosh \left (a+b \sqrt{c+d x}\right )}{b^4 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*Sqrt[c + d*x]],x]

[Out]

(-2*(6 + b^2*(2*c + 3*d*x))*Cosh[a + b*Sqrt[c + d*x]] + 2*b*Sqrt[c + d*x]*(6 + b^2*d*x)*Sinh[a + b*Sqrt[c + d*
x]])/(b^4*d^2)

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Maple [B]  time = 0.013, size = 303, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{{d}^{2}{b}^{2}} \left ({\frac{ \left ( a+b\sqrt{dx+c} \right ) ^{3}\sinh \left ( a+b\sqrt{dx+c} \right ) -3\, \left ( a+b\sqrt{dx+c} \right ) ^{2}\cosh \left ( a+b\sqrt{dx+c} \right ) +6\, \left ( a+b\sqrt{dx+c} \right ) \sinh \left ( a+b\sqrt{dx+c} \right ) -6\,\cosh \left ( a+b\sqrt{dx+c} \right ) }{{b}^{2}}}-3\,{\frac{a \left ( \left ( a+b\sqrt{dx+c} \right ) ^{2}\sinh \left ( a+b\sqrt{dx+c} \right ) -2\, \left ( a+b\sqrt{dx+c} \right ) \cosh \left ( a+b\sqrt{dx+c} \right ) +2\,\sinh \left ( a+b\sqrt{dx+c} \right ) \right ) }{{b}^{2}}}+3\,{\frac{{a}^{2} \left ( \left ( a+b\sqrt{dx+c} \right ) \sinh \left ( a+b\sqrt{dx+c} \right ) -\cosh \left ( a+b\sqrt{dx+c} \right ) \right ) }{{b}^{2}}}-{\frac{{a}^{3}\sinh \left ( a+b\sqrt{dx+c} \right ) }{{b}^{2}}}-c \left ( \left ( a+b\sqrt{dx+c} \right ) \sinh \left ( a+b\sqrt{dx+c} \right ) -\cosh \left ( a+b\sqrt{dx+c} \right ) \right ) +ac\sinh \left ( a+b\sqrt{dx+c} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(a+b*(d*x+c)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.15458, size = 393, normalized size = 2.35 \begin{align*} \frac{2 \, d^{2} x^{2} \cosh \left (\sqrt{d x + c} b + a\right ) -{\left (\frac{c^{2} e^{\left (\sqrt{d x + c} b + a\right )}}{b} + \frac{c^{2} e^{\left (-\sqrt{d x + c} b - a\right )}}{b} - \frac{2 \,{\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt{d x + c} b e^{a} + 2 \, e^{a}\right )} c e^{\left (\sqrt{d x + c} b\right )}}{b^{3}} - \frac{2 \,{\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt{d x + c} b + 2\right )} c e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{3}} + \frac{{\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} e^{a} + 12 \,{\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt{d x + c} b e^{a} + 24 \, e^{a}\right )} e^{\left (\sqrt{d x + c} b\right )}}{b^{5}} + \frac{{\left ({\left (d x + c\right )}^{2} b^{4} + 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} + 12 \,{\left (d x + c\right )} b^{2} + 24 \, \sqrt{d x + c} b + 24\right )} e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{5}}\right )} b}{4 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

1/4*(2*d^2*x^2*cosh(sqrt(d*x + c)*b + a) - (c^2*e^(sqrt(d*x + c)*b + a)/b + c^2*e^(-sqrt(d*x + c)*b - a)/b - 2
*((d*x + c)*b^2*e^a - 2*sqrt(d*x + c)*b*e^a + 2*e^a)*c*e^(sqrt(d*x + c)*b)/b^3 - 2*((d*x + c)*b^2 + 2*sqrt(d*x
+ c)*b + 2)*c*e^(-sqrt(d*x + c)*b - a)/b^3 + ((d*x + c)^2*b^4*e^a - 4*(d*x + c)^(3/2)*b^3*e^a + 12*(d*x + c)*
b^2*e^a - 24*sqrt(d*x + c)*b*e^a + 24*e^a)*e^(sqrt(d*x + c)*b)/b^5 + ((d*x + c)^2*b^4 + 4*(d*x + c)^(3/2)*b^3
+ 12*(d*x + c)*b^2 + 24*sqrt(d*x + c)*b + 24)*e^(-sqrt(d*x + c)*b - a)/b^5)*b)/d^2

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Fricas [A]  time = 1.80244, size = 169, normalized size = 1.01 \begin{align*} \frac{2 \,{\left ({\left (b^{3} d x + 6 \, b\right )} \sqrt{d x + c} \sinh \left (\sqrt{d x + c} b + a\right ) -{\left (3 \, b^{2} d x + 2 \, b^{2} c + 6\right )} \cosh \left (\sqrt{d x + c} b + a\right )\right )}}{b^{4} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

2*((b^3*d*x + 6*b)*sqrt(d*x + c)*sinh(sqrt(d*x + c)*b + a) - (3*b^2*d*x + 2*b^2*c + 6)*cosh(sqrt(d*x + c)*b +
a))/(b^4*d^2)

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Sympy [A]  time = 0.713937, size = 151, normalized size = 0.9 \begin{align*} \begin{cases} \frac{x^{2} \cosh{\left (a \right )}}{2} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x^{2} \cosh{\left (a + b \sqrt{c} \right )}}{2} & \text{for}\: d = 0 \\\frac{2 x \sqrt{c + d x} \sinh{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{4 c \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} - \frac{6 x \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{12 \sqrt{c + d x} \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} - \frac{12 \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(a+b*(d*x+c)**(1/2)),x)

[Out]

Piecewise((x**2*cosh(a)/2, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**2*cosh(a + b*sqrt(c))/2, Eq(d, 0)), (2*x*sqr
t(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b*d) - 4*c*cosh(a + b*sqrt(c + d*x))/(b**2*d**2) - 6*x*cosh(a + b*sqrt(c
+ d*x))/(b**2*d) + 12*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b**3*d**2) - 12*cosh(a + b*sqrt(c + d*x))/(b**
4*d**2), True))

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Giac [B]  time = 1.50521, size = 782, normalized size = 4.68 \begin{align*} -\frac{\frac{{\left ({\left (\sqrt{d x + c} b + a\right )} b^{2} c - a b^{2} c - b^{2} c \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) -{\left (\sqrt{d x + c} b + a\right )}^{3} + 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} a - 3 \,{\left (\sqrt{d x + c} b + a\right )} a^{2} + a^{3} + 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 6 \,{\left (\sqrt{d x + c} b + a\right )} a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 3 \, a^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 6 \, \sqrt{d x + c} b + 6 \, \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )} e^{\left ({\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a\right )}}{b^{3} d} - \frac{{\left ({\left (\sqrt{d x + c} b + a\right )} b^{2} c - a b^{2} c + b^{2} c \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) -{\left (\sqrt{d x + c} b + a\right )}^{3} + 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} a - 3 \,{\left (\sqrt{d x + c} b + a\right )} a^{2} + a^{3} - 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 6 \,{\left (\sqrt{d x + c} b + a\right )} a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \, a^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 6 \, \sqrt{d x + c} b - 6 \, \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )} e^{\left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right )}}{b^{3} d}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

-(((sqrt(d*x + c)*b + a)*b^2*c - a*b^2*c - b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - (sqrt(d*x + c)*b + a)^3
+ 3*(sqrt(d*x + c)*b + a)^2*a - 3*(sqrt(d*x + c)*b + a)*a^2 + a^3 + 3*(sqrt(d*x + c)*b + a)^2*sgn((sqrt(d*x +
c)*b + a)*b - a*b) - 6*(sqrt(d*x + c)*b + a)*a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 3*a^2*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) - 6*sqrt(d*x + c)*b + 6*sgn((sqrt(d*x + c)*b + a)*b - a*b))*e^((sqrt(d*x + c)*b + a)*sgn((sqrt(
d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)/(b^3*d) - ((sqrt(d*x + c)*b + a)*b^2*c -
a*b^2*c + b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - (sqrt(d*x + c)*b + a)^3 + 3*(sqrt(d*x + c)*b + a)^2*a - 3
*(sqrt(d*x + c)*b + a)*a^2 + a^3 - 3*(sqrt(d*x + c)*b + a)^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 6*(sqrt(d*x
+ c)*b + a)*a*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 6*sqrt(d*x + c)*
b - 6*sgn((sqrt(d*x + c)*b + a)*b - a*b))*e^(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn
((sqrt(d*x + c)*b + a)*b - a*b) - a)/(b^3*d))/(b*d)