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

Optimal. Leaf size=123 \[ -\frac{\sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \cosh (a+b x)}{b} \]

[Out]

(Sqrt[c + d*x]*Cosh[a + b*x])/b - (Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*
b^(3/2)) - (Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))

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Rubi [A]  time = 0.187407, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3296, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]*Sinh[a + b*x],x]

[Out]

(Sqrt[c + d*x]*Cosh[a + b*x])/b - (Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*
b^(3/2)) - (Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))

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 3307

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

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 \sqrt{c+d x} \sinh (a+b x) \, dx &=\frac{\sqrt{c+d x} \cosh (a+b x)}{b}-\frac{d \int \frac{\cosh (a+b x)}{\sqrt{c+d x}} \, dx}{2 b}\\ &=\frac{\sqrt{c+d x} \cosh (a+b x)}{b}-\frac{d \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b}-\frac{d \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b}\\ &=\frac{\sqrt{c+d x} \cosh (a+b x)}{b}-\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 )}{2 b}-\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 )}{2 b}\\ &=\frac{\sqrt{c+d x} \cosh (a+b x)}{b}-\frac{\sqrt{d} e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{d} e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0779187, size = 104, normalized size = 0.85 \[ \frac{\sqrt{c+d x} e^{-a-\frac{b c}{d}} \left (\frac{e^{2 a} \text{Gamma}\left (\frac{3}{2},-\frac{b (c+d x)}{d}\right )}{\sqrt{-\frac{b (c+d x)}{d}}}+\frac{e^{\frac{2 b c}{d}} \text{Gamma}\left (\frac{3}{2},\frac{b (c+d x)}{d}\right )}{\sqrt{\frac{b (c+d x)}{d}}}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-a - (b*c)/d)*Sqrt[c + d*x]*((E^(2*a)*Gamma[3/2, -((b*(c + d*x))/d)])/Sqrt[-((b*(c + d*x))/d)] + (E^((2*b*
c)/d)*Gamma[3/2, (b*(c + d*x))/d])/Sqrt[(b*(c + d*x))/d]))/(2*b)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int \sinh \left ( bx+a \right ) \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 = 1.15584, size = 311, normalized size = 2.53 \begin{align*} \frac{8 \,{\left (d x + c\right )}^{\frac{3}{2}} \sinh \left (b x + a\right ) - \frac{{\left (\frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b^{2} \sqrt{-\frac{b}{d}}} + \frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b^{2} \sqrt{\frac{b}{d}}} - \frac{2 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (\frac{b c}{d}\right )} + 3 \, \sqrt{d x + c} d^{2} e^{\left (\frac{b c}{d}\right )}\right )} e^{\left (-a - \frac{{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac{2 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{a} - 3 \, \sqrt{d x + c} d^{2} e^{a}\right )} e^{\left (\frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b^{2}}\right )} b}{d}}{12 \, 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]

1/12*(8*(d*x + c)^(3/2)*sinh(b*x + a) - (3*sqrt(pi)*d^2*erf(sqrt(d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/(b^2*sqrt(
-b/d)) + 3*sqrt(pi)*d^2*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b^2*sqrt(b/d)) - 2*(2*(d*x + c)^(3/2)*b*d
*e^(b*c/d) + 3*sqrt(d*x + c)*d^2*e^(b*c/d))*e^(-a - (d*x + c)*b/d)/b^2 + 2*(2*(d*x + c)^(3/2)*b*d*e^a - 3*sqrt
(d*x + c)*d^2*e^a)*e^((d*x + c)*b/d - b*c/d)/b^2)*b/d)/d

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Fricas [B]  time = 2.73369, size = 718, normalized size = 5.84 \begin{align*} -\frac{\sqrt{\pi }{\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac{b c - a d}{d}\right ) - d \cosh \left (b x + a\right ) \sinh \left (-\frac{b c - a d}{d}\right ) +{\left (d \cosh \left (-\frac{b c - a d}{d}\right ) - d \sinh \left (-\frac{b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt{\frac{b}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) - \sqrt{\pi }{\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac{b c - a d}{d}\right ) + d \cosh \left (b x + a\right ) \sinh \left (-\frac{b c - a d}{d}\right ) +{\left (d \cosh \left (-\frac{b c - a d}{d}\right ) + d \sinh \left (-\frac{b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt{-\frac{b}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) - 2 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b\right )} \sqrt{d x + c}}{4 \,{\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )}} \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]

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

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

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Giac [A]  time = 1.27523, size = 227, normalized size = 1.85 \begin{align*} \frac{\frac{\sqrt{\pi } d^{2} \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} b} + \frac{\sqrt{\pi } d^{2} \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} b} + \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b} + \frac{2 \, \sqrt{d x + c} d e^{\left (-\frac{{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b}}{4 \, 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]

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