3.63 \(\int \frac{\sinh (f x)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{\sqrt{\pi } \sqrt{f} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{\sqrt{\pi } \sqrt{f} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \sinh (f x)}{d \sqrt{d x}} \]

[Out]

(Sqrt[f]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[f]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[
d]])/d^(3/2) - (2*Sinh[f*x])/(d*Sqrt[d*x])

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Rubi [A]  time = 0.112284, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3297, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \sqrt{f} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{\sqrt{\pi } \sqrt{f} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \sinh (f x)}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[f*x]/(d*x)^(3/2),x]

[Out]

(Sqrt[f]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[f]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[
d]])/d^(3/2) - (2*Sinh[f*x])/(d*Sqrt[d*x])

Rule 3297

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

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

Mathematica [A]  time = 0.0198243, size = 49, normalized size = 0.56 \[ \frac{x \left (\sqrt{-f x} \text{Gamma}\left (\frac{1}{2},-f x\right )-\sqrt{f x} \text{Gamma}\left (\frac{1}{2},f x\right )-2 \sinh (f x)\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[f*x]/(d*x)^(3/2),x]

[Out]

(x*(Sqrt[-(f*x)]*Gamma[1/2, -(f*x)] - Sqrt[f*x]*Gamma[1/2, f*x] - 2*Sinh[f*x]))/(d*x)^(3/2)

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Maple [C]  time = 0.022, size = 120, normalized size = 1.4 \begin{align*} -{\frac{\sqrt{\pi }\sqrt{2}}{4\,f}{x}^{{\frac{3}{2}}} \left ( if \right ) ^{{\frac{3}{2}}} \left ( 2\,{\frac{\sqrt{2}\sqrt{if}{{\rm e}^{-fx}}}{\sqrt{\pi }\sqrt{x}f}}-2\,{\frac{\sqrt{2}\sqrt{if}{{\rm e}^{fx}}}{\sqrt{\pi }\sqrt{x}f}}+2\,{\frac{\sqrt{2}\sqrt{if}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ) }{\sqrt{f}}}+2\,{\frac{\sqrt{2}\sqrt{if}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ) }{\sqrt{f}}} \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(f*x)/(d*x)^(3/2),x)

[Out]

-1/4*Pi^(1/2)/(d*x)^(3/2)*x^(3/2)*2^(1/2)*(I*f)^(3/2)/f*(2/Pi^(1/2)/x^(1/2)*2^(1/2)*(I*f)^(1/2)/f*exp(-f*x)-2/
Pi^(1/2)/x^(1/2)*2^(1/2)*(I*f)^(1/2)/f*exp(f*x)+2*(I*f)^(1/2)*2^(1/2)/f^(1/2)*erf(x^(1/2)*f^(1/2))+2*(I*f)^(1/
2)*2^(1/2)/f^(1/2)*erfi(x^(1/2)*f^(1/2)))

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Maxima [A]  time = 1.13607, size = 100, normalized size = 1.15 \begin{align*} \frac{\frac{f{\left (\frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right )}{\sqrt{\frac{f}{d}}} + \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{\sqrt{-\frac{f}{d}}}\right )}}{d} - \frac{2 \, \sinh \left (f x\right )}{\sqrt{d x}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

(f*(sqrt(pi)*erf(sqrt(d*x)*sqrt(f/d))/sqrt(f/d) + sqrt(pi)*erf(sqrt(d*x)*sqrt(-f/d))/sqrt(-f/d))/d - 2*sinh(f*
x)/sqrt(d*x))/d

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Fricas [B]  time = 2.69863, size = 355, normalized size = 4.08 \begin{align*} \frac{\sqrt{\pi }{\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt{\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right ) - \sqrt{\pi }{\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt{-\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right ) - \sqrt{d x}{\left (\cosh \left (f x\right )^{2} + 2 \, \cosh \left (f x\right ) \sinh \left (f x\right ) + \sinh \left (f x\right )^{2} - 1\right )}}{d^{2} x \cosh \left (f x\right ) + d^{2} x \sinh \left (f x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

(sqrt(pi)*(d*x*cosh(f*x) + d*x*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) - sqrt(pi)*(d*x*cosh(f*x) + d*x*s
inh(f*x))*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)) - sqrt(d*x)*(cosh(f*x)^2 + 2*cosh(f*x)*sinh(f*x) + sinh(f*x)^2
- 1))/(d^2*x*cosh(f*x) + d^2*x*sinh(f*x))

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Sympy [C]  time = 7.42422, size = 94, normalized size = 1.08 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } \sqrt{f} e^{- \frac{i \pi }{4}} C\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x} e^{\frac{i \pi }{4}}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{2 d^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} - \frac{\sinh{\left (f x \right )} \Gamma \left (\frac{1}{4}\right )}{2 d^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x)/(d*x)**(3/2),x)

[Out]

sqrt(2)*sqrt(pi)*sqrt(f)*exp(-I*pi/4)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(1/4)/(2*d**
(3/2)*gamma(5/4)) - sinh(f*x)*gamma(1/4)/(2*d**(3/2)*sqrt(x)*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (f x\right )}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x)/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(f*x)/(d*x)^(3/2), x)