3.1.41 \(\int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx\) [41]

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

[Out]

-1/2*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(1/2)/d^(1/2)+1/2*exp(a-b*c/d)*erfi(b^(1/2)*(
d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(1/2)/d^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3389, 2211, 2235, 2236} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]/Sqrt[c + d*x],x]

[Out]

-1/2*(E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqr
t[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d])

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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]

Rule 3389

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]

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 {\text {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 {\text {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 = 0.02, size = 104, normalized size = 1.00 \begin {gather*} \frac {e^{-a-\frac {b c}{d}} \left (e^{2 a} \sqrt {-\frac {b (c+d x)}{d}} \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}} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )}{2 b \sqrt {c+d x}} \end {gather*}

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])

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh \left (b x +a \right )}{\sqrt {d x +c}}\, dx\]

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)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (74) = 148\).
time = 0.28, size = 181, normalized size = 1.74 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 122, normalized size = 1.17 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.44, size = 90, normalized size = 0.87 \begin {gather*} \frac {{\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c}{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 - 2 \, a d}{d}\right )}}{\sqrt {-b d}}\right )} e^{\left (-a\right )}}{2 \, d} \end {gather*}

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x\right )}{\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)/(c + d*x)^(1/2),x)

[Out]

int(sinh(a + b*x)/(c + d*x)^(1/2), x)

________________________________________________________________________________________