3.95 \(\int \sinh (a+b \sqrt{c+d x}) \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d}-\frac{2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0457324, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5310, 5304, 3296, 2637} \[ \frac{2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d}-\frac{2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Sqrt[c + d*x]],x]

[Out]

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

Rule 5310

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

Rule 5304

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Module[{k = Denominator[n]}, Dist[k, Sub
st[Int[x^(k - 1)*(a + b*Sinh[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && FractionQ[n]
&& IntegerQ[p]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0626198, size = 50, normalized size = 0.93 \[ \frac{2 \left (b \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )-\sinh \left (a+b \sqrt{c+d x}\right )\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 63, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( a+b\sqrt{dx+c} \right ) \cosh \left ( a+b\sqrt{dx+c} \right ) -\sinh \left ( a+b\sqrt{dx+c} \right ) -a\cosh \left ( a+b\sqrt{dx+c} \right ) }{d{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.14806, size = 150, normalized size = 2.78 \begin{align*} -\frac{b{\left (\frac{{\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt{d x + c} b e^{a} + 2 \, e^{a}\right )} e^{\left (\sqrt{d x + c} b\right )}}{b^{3}} - \frac{{\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt{d x + c} b + 2\right )} e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{3}}\right )} - 2 \,{\left (d x + c\right )} \sinh \left (\sqrt{d x + c} b + a\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.03717, size = 112, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (\sqrt{d x + c} b \cosh \left (\sqrt{d x + c} b + a\right ) - \sinh \left (\sqrt{d x + c} b + a\right )\right )}}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.628374, size = 65, normalized size = 1.2 \begin{align*} \begin{cases} x \sinh{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sinh{\left (a + b \sqrt{c} \right )} & \text{for}\: d = 0 \\\frac{2 \sqrt{c + d x} \cosh{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{2 \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a), Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x*sinh(a + b*sqrt(c)), Eq(d, 0)), (2*sqrt(c + d*x)*c
osh(a + b*sqrt(c + d*x))/(b*d) - 2*sinh(a + b*sqrt(c + d*x))/(b**2*d), True))

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Giac [B]  time = 1.32425, size = 281, normalized size = 5.2 \begin{align*} \frac{{\left ({\left (\sqrt{d x + c} b + a\right )} b - a b - b \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 - a b + b \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((sqrt(d*x + c)*b + a)*b - a*b - b*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 - a*b + b*
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)