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

Optimal. Leaf size=104 \[ \frac{\sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}-\frac{\sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

[Out]

-(E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqrt[
Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d])

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Rubi [A]  time = 0.134614, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}-\frac{\sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]/Sqrt[c + d*x],x]

[Out]

-(E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqrt[
Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d])

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
 2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sinh (a+b x)}{\sqrt{c+d x}} \, dx &=\frac{1}{2} \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx-\frac{1}{2} \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int e^{i \left (i a-\frac{i b c}{d}\right )-\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}+\frac{\operatorname{Subst}\left (\int e^{-i \left (i a-\frac{i b c}{d}\right )+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.0376037, size = 104, normalized size = 1. \[ \frac{e^{-a-\frac{b c}{d}} \left (e^{2 a} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{b (c+d x)}{d}\right )+e^{\frac{2 b c}{d}} \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},\frac{b (c+d x)}{d}\right )\right )}{2 b \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-a - (b*c)/d)*(E^(2*a)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)] + E^((2*b*c)/d)*Sqrt[(b*(c
+ d*x))/d]*Gamma[1/2, (b*(c + d*x))/d]))/(2*b*Sqrt[c + d*x])

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\sinh \left ( bx+a \right ){\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15108, size = 244, normalized size = 2.35 \begin{align*} \frac{4 \, \sqrt{d x + c} \sinh \left (b x + a\right ) + \frac{{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b \sqrt{-\frac{b}{d}}} - \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b \sqrt{\frac{b}{d}}} - \frac{2 \, \sqrt{d x + c} d e^{\left (a + \frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b} + \frac{2 \, \sqrt{d x + c} d e^{\left (-a - \frac{{\left (d x + c\right )} b}{d} + \frac{b c}{d}\right )}}{b}\right )} b}{d}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*sqrt(d*x + c)*sinh(b*x + a) + (sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/(b*sqrt(-b/d)) -
sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b*sqrt(b/d)) - 2*sqrt(d*x + c)*d*e^(a + (d*x + c)*b/d
- b*c/d)/b + 2*sqrt(d*x + c)*d*e^(-a - (d*x + c)*b/d + b*c/d)/b)*b/d)/d

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Fricas [A]  time = 2.62937, size = 273, normalized size = 2.62 \begin{align*} -\frac{\sqrt{\pi } \sqrt{\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) - \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) + \sqrt{\pi } \sqrt{-\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) + \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(pi)*sqrt(b/d)*(cosh(-(b*c - a*d)/d) - sinh(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(b/d)) + sqrt(pi)
*sqrt(-b/d)*(cosh(-(b*c - a*d)/d) + sinh(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-b/d)))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x \right )}}{\sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x)/sqrt(c + d*x), x)

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Giac [A]  time = 1.21713, size = 124, normalized size = 1.19 \begin{align*} \frac{\frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}}{d}\right ) e^{\left (\frac{b c - a d}{d}\right )}}{\sqrt{b d}} - \frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{-b d} \sqrt{d x + c}}{d}\right ) e^{\left (-\frac{b c - a d}{d}\right )}}{\sqrt{-b d}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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