\(\int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx\) [64]

Optimal result

Integrand size = 12, antiderivative size = 114 \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 f^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 f^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}} \]

[Out]

-2/3*sinh(f*x)/d/(d*x)^(3/2)-2/3*f^(3/2)*erf(f^(1/2)*(d*x)^(1/2)/d^(1/2))*Pi^(1/2)/d^(5/2)+2/3*f^(3/2)*erfi(f^
(1/2)*(d*x)^(1/2)/d^(1/2))*Pi^(1/2)/d^(5/2)-4/3*f*cosh(f*x)/d^2/(d*x)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3378, 3389, 2211, 2235, 2236} \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=-\frac {2 \sqrt {\pi } f^{3/2} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {\pi } f^{3/2} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}} \]

[In]

Int[Sinh[f*x]/(d*x)^(5/2),x]

[Out]

(-4*f*Cosh[f*x])/(3*d^2*Sqrt[d*x]) - (2*f^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*f^
(3/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*Sinh[f*x])/(3*d*(d*x)^(3/2))

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 3378

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 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*} \text {integral}& = -\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}+\frac {(2 f) \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx}{3 d} \\ & = -\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}+\frac {\left (4 f^2\right ) \int \frac {\sinh (f x)}{\sqrt {d x}} \, dx}{3 d^2} \\ & = -\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}-\frac {\left (2 f^2\right ) \int \frac {e^{-f x}}{\sqrt {d x}} \, dx}{3 d^2}+\frac {\left (2 f^2\right ) \int \frac {e^{f x}}{\sqrt {d x}} \, dx}{3 d^2} \\ & = -\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}-\frac {\left (4 f^2\right ) \text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{3 d^3}+\frac {\left (4 f^2\right ) \text {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{3 d^3} \\ & = -\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 f^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 f^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=-\frac {e^{-f x} x \left (-1+e^{2 f x}+2 f x+2 e^{2 f x} f x+2 e^{f x} (-f x)^{3/2} \Gamma \left (\frac {1}{2},-f x\right )-2 e^{f x} (f x)^{3/2} \Gamma \left (\frac {1}{2},f x\right )\right )}{3 (d x)^{5/2}} \]

[In]

Integrate[Sinh[f*x]/(d*x)^(5/2),x]

[Out]

-1/3*(x*(-1 + E^(2*f*x) + 2*f*x + 2*E^(2*f*x)*f*x + 2*E^(f*x)*(-(f*x))^(3/2)*Gamma[1/2, -(f*x)] - 2*E^(f*x)*(f
*x)^(3/2)*Gamma[1/2, f*x]))/(E^(f*x)*(d*x)^(5/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.16

method	result	size
meijerg	\(-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (-\frac {4 \sqrt {2}\, \left (2 f x +1\right ) {\mathrm e}^{f x}}{3 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {i f}\, f}+\frac {4 \sqrt {2}\, \left (-2 f x +1\right ) {\mathrm e}^{-f x}}{3 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {i f}\, f}-\frac {8 \sqrt {2}\, \sqrt {f}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{3 \sqrt {i f}}+\frac {8 \sqrt {2}\, \sqrt {f}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{3 \sqrt {i f}}\right )}{8 \left (d x \right )^{\frac {5}{2}} f}\)	\(132\)

[In]

int(sinh(f*x)/(d*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/8*Pi^(1/2)/(d*x)^(5/2)*x^(5/2)*2^(1/2)*(I*f)^(5/2)/f*(-4/3/Pi^(1/2)/x^(3/2)*2^(1/2)/(I*f)^(1/2)*(2*f*x+1)/f
*exp(f*x)+4/3/Pi^(1/2)/x^(3/2)*2^(1/2)/(I*f)^(1/2)*(-2*f*x+1)/f*exp(-f*x)-8/3/(I*f)^(1/2)*2^(1/2)*f^(1/2)*erf(
x^(1/2)*f^(1/2))+8/3/(I*f)^(1/2)*2^(1/2)*f^(1/2)*erfi(x^(1/2)*f^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (78) = 156\).

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.56 \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {\pi } {\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + 2 \, \sqrt {\pi } {\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + {\left ({\left (2 \, f x + 1\right )} \cosh \left (f x\right )^{2} + 2 \, {\left (2 \, f x + 1\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) + {\left (2 \, f x + 1\right )} \sinh \left (f x\right )^{2} + 2 \, f x - 1\right )} \sqrt {d x}}{3 \, {\left (d^{3} x^{2} \cosh \left (f x\right ) + d^{3} x^{2} \sinh \left (f x\right )\right )}} \]

[In]

integrate(sinh(f*x)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*sqrt(pi)*(d*f*x^2*cosh(f*x) + d*f*x^2*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) + 2*sqrt(pi)*(d*f*
x^2*cosh(f*x) + d*f*x^2*sinh(f*x))*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)) + ((2*f*x + 1)*cosh(f*x)^2 + 2*(2*f*x
+ 1)*cosh(f*x)*sinh(f*x) + (2*f*x + 1)*sinh(f*x)^2 + 2*f*x - 1)*sqrt(d*x))/(d^3*x^2*cosh(f*x) + d^3*x^2*sinh(f
*x))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=- \frac {\sqrt {2} \sqrt {\pi } f^{\frac {3}{2}} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {f \cosh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {\sinh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{6 d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]

[In]

integrate(sinh(f*x)/(d*x)**(5/2),x)

[Out]

-sqrt(2)*sqrt(pi)*f**(3/2)*exp(-3*I*pi/4)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(-1/4)/(
3*d**(5/2)*gamma(3/4)) + f*cosh(f*x)*gamma(-1/4)/(3*d**(5/2)*sqrt(x)*gamma(3/4)) + sinh(f*x)*gamma(-1/4)/(6*d*
*(5/2)*x**(3/2)*gamma(3/4))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50 \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=-\frac {\frac {f {\left (\frac {\sqrt {f x} \Gamma \left (-\frac {1}{2}, f x\right )}{\sqrt {d x}} + \frac {\sqrt {-f x} \Gamma \left (-\frac {1}{2}, -f x\right )}{\sqrt {d x}}\right )}}{d} + \frac {2 \, \sinh \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}}}{3 \, d} \]

[In]

integrate(sinh(f*x)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(f*(sqrt(f*x)*gamma(-1/2, f*x)/sqrt(d*x) + sqrt(-f*x)*gamma(-1/2, -f*x)/sqrt(d*x))/d + 2*sinh(f*x)/(d*x)^
(3/2))/d

Giac [F]

\[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=\int { \frac {\sinh \left (f x\right )}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(sinh(f*x)/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate(sinh(f*x)/(d*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx=\int \frac {\mathrm {sinh}\left (f\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]

[In]

int(sinh(f*x)/(d*x)^(5/2),x)

[Out]

int(sinh(f*x)/(d*x)^(5/2), x)