3.1.60 \(\int (d x)^{3/2} \sinh (f x) \, dx\) [60]

Optimal. Leaf size=111 \[ \frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {3 d^{3/2} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 d^{3/2} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2} \]

[Out]

(d*x)^(3/2)*cosh(f*x)/f-3/8*d^(3/2)*erf(f^(1/2)*(d*x)^(1/2)/d^(1/2))*Pi^(1/2)/f^(5/2)+3/8*d^(3/2)*erfi(f^(1/2)
*(d*x)^(1/2)/d^(1/2))*Pi^(1/2)/f^(5/2)-3/2*d*sinh(f*x)*(d*x)^(1/2)/f^2

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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3377, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {3 \sqrt {\pi } d^{3/2} \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 \sqrt {\pi } d^{3/2} \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2}+\frac {(d x)^{3/2} \cosh (f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*x)^(3/2)*Cosh[f*x])/f - (3*d^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) + (3*d^(3/2)*Sqr
t[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) - (3*d*Sqrt[d*x]*Sinh[f*x])/(2*f^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 3377

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 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 (d x)^{3/2} \sinh (f x) \, dx &=\frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {(3 d) \int \sqrt {d x} \cosh (f x) \, dx}{2 f}\\ &=\frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2}+\frac {\left (3 d^2\right ) \int \frac {\sinh (f x)}{\sqrt {d x}} \, dx}{4 f^2}\\ &=\frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2}-\frac {\left (3 d^2\right ) \int \frac {e^{-f x}}{\sqrt {d x}} \, dx}{8 f^2}+\frac {\left (3 d^2\right ) \int \frac {e^{f x}}{\sqrt {d x}} \, dx}{8 f^2}\\ &=\frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2}-\frac {(3 d) \text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{4 f^2}+\frac {(3 d) \text {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{4 f^2}\\ &=\frac {(d x)^{3/2} \cosh (f x)}{f}-\frac {3 d^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 d^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}-\frac {3 d \sqrt {d x} \sinh (f x)}{2 f^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 50, normalized size = 0.45 \begin {gather*} \frac {d^2 \left (\sqrt {-f x} \Gamma \left (\frac {5}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {5}{2},f x\right )\right )}{2 f^3 \sqrt {d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.24, size = 132, normalized size = 1.19

method	result	size
meijerg	\(-\frac {2 \left (d x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {7}{2}} \left (-14 f x +21\right ) {\mathrm e}^{f x}}{112 \sqrt {\pi }\, f^{3}}+\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {7}{2}} \left (14 f x +21\right ) {\mathrm e}^{-f x}}{112 \sqrt {\pi }\, f^{3}}-\frac {3 \left (i f \right )^{\frac {7}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {7}{2}}}+\frac {3 \left (i f \right )^{\frac {7}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {7}{2}}}\right )}{x^{\frac {3}{2}} \left (i f \right )^{\frac {3}{2}} f}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*x)^(3/2)/x^(3/2)*2^(1/2)/(I*f)^(3/2)*Pi^(1/2)/f*(-1/112/Pi^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(7/2)*(-14*f*x+21
)/f^3*exp(f*x)+1/112/Pi^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(7/2)*(14*f*x+21)/f^3*exp(-f*x)-3/32*(I*f)^(7/2)*2^(1/2)/f
^(7/2)*erf(x^(1/2)*f^(1/2))+3/32*(I*f)^(7/2)*2^(1/2)/f^(7/2)*erfi(x^(1/2)*f^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (77) = 154\).
time = 0.26, size = 175, normalized size = 1.58 \begin {gather*} \frac {16 \, \left (d x\right )^{\frac {5}{2}} \sinh \left (f x\right ) - \frac {f {\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{3} \sqrt {\frac {f}{d}}} - \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{3} \sqrt {-\frac {f}{d}}} + \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} - 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} + 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (-f x\right )}}{f^{3}}\right )}}{d}}{40 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(16*(d*x)^(5/2)*sinh(f*x) - f*(15*sqrt(pi)*d^3*erf(sqrt(d*x)*sqrt(f/d))/(f^3*sqrt(f/d)) - 15*sqrt(pi)*d^3
*erf(sqrt(d*x)*sqrt(-f/d))/(f^3*sqrt(-f/d)) + 2*(4*(d*x)^(5/2)*d*f^2 - 10*(d*x)^(3/2)*d^2*f + 15*sqrt(d*x)*d^3
)*e^(f*x)/f^3 - 2*(4*(d*x)^(5/2)*d*f^2 + 10*(d*x)^(3/2)*d^2*f + 15*sqrt(d*x)*d^3)*e^(-f*x)/f^3)/d)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (77) = 154\).
time = 0.34, size = 189, normalized size = 1.70 \begin {gather*} -\frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - 2 \, {\left (2 \, d f^{2} x + {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right )^{2} + 2 \, {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) + {\left (2 \, d f^{2} x - 3 \, d f\right )} \sinh \left (f x\right )^{2} + 3 \, d f\right )} \sqrt {d x}}{8 \, {\left (f^{3} \cosh \left (f x\right ) + f^{3} \sinh \left (f x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*sqrt(pi)*(d^2*cosh(f*x) + d^2*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) + 3*sqrt(pi)*(d^2*cosh(f*x
) + d^2*sinh(f*x))*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)) - 2*(2*d*f^2*x + (2*d*f^2*x - 3*d*f)*cosh(f*x)^2 + 2*(
2*d*f^2*x - 3*d*f)*cosh(f*x)*sinh(f*x) + (2*d*f^2*x - 3*d*f)*sinh(f*x)^2 + 3*d*f)*sqrt(d*x))/(f^3*cosh(f*x) +
f^3*sinh(f*x))

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Sympy [C] Result contains complex when optimal does not.
time = 13.77, size = 133, normalized size = 1.20 \begin {gather*} \frac {7 d^{\frac {3}{2}} x^{\frac {3}{2}} \cosh {\left (f x \right )} \Gamma \left (\frac {7}{4}\right )}{4 f \Gamma \left (\frac {11}{4}\right )} - \frac {21 d^{\frac {3}{2}} \sqrt {x} \sinh {\left (f x \right )} \Gamma \left (\frac {7}{4}\right )}{8 f^{2} \Gamma \left (\frac {11}{4}\right )} + \frac {21 \sqrt {2} \sqrt {\pi } d^{\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 {7}{4}\right )}{16 f^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*d**(3/2)*x**(3/2)*cosh(f*x)*gamma(7/4)/(4*f*gamma(11/4)) - 21*d**(3/2)*sqrt(x)*sinh(f*x)*gamma(7/4)/(8*f**2*
gamma(11/4)) + 21*sqrt(2)*sqrt(pi)*d**(3/2)*exp(-3*I*pi/4)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(p
i))*gamma(7/4)/(16*f**(5/2)*gamma(11/4))

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Giac [A]
time = 0.44, size = 146, normalized size = 1.32 \begin {gather*} \frac {1}{8} \, d {\left (\frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f^{2}} + \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}}{d^{2}} - \frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f^{2}} - \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}}}{d^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d*((3*sqrt(pi)*d^3*erf(-sqrt(d*f)*sqrt(d*x)/d)/(sqrt(d*f)*f^2) + 2*(2*sqrt(d*x)*d^2*f*x + 3*sqrt(d*x)*d^2)
*e^(-f*x)/f^2)/d^2 - (3*sqrt(pi)*d^3*erf(-sqrt(-d*f)*sqrt(d*x)/d)/(sqrt(-d*f)*f^2) - 2*(2*sqrt(d*x)*d^2*f*x -
3*sqrt(d*x)*d^2)*e^(f*x)/f^2)/d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {sinh}\left (f\,x\right )\,{\left (d\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinh(f*x)*(d*x)^(3/2), x)

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