Integrand size = 12, antiderivative size = 111 \[ \int (d x)^{3/2} \cosh (f x) \, dx=-\frac {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {3 d^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 d^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {(d x)^{3/2} \sinh (f x)}{f} \]
(d*x)^(3/2)*sinh(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*cosh(f*x)*(d*x)^(1/2)/f^2
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.46 \[ \int (d x)^{3/2} \cosh (f x) \, dx=\frac {d^2 \left (\sqrt {-f x} \Gamma \left (\frac {5}{2},-f x\right )-\sqrt {f x} \Gamma \left (\frac {5}{2},f x\right )\right )}{2 f^3 \sqrt {d x}} \]
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3777, 26, 3042, 26, 3777, 3042, 3788, 26, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d x)^{3/2} \cosh (f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d x)^{3/2} \sin \left (\frac {\pi }{2}+i f x\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}-\frac {3 i d \int -i \sqrt {d x} \sinh (f x)dx}{2 f}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}-\frac {3 d \int \sqrt {d x} \sinh (f x)dx}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}-\frac {3 d \int -i \sqrt {d x} \sin (i f x)dx}{2 f}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \int \sqrt {d x} \sin (i f x)dx}{2 f}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \int \frac {\cosh (f x)}{\sqrt {d x}}dx}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \int \frac {\sin \left (i f x+\frac {\pi }{2}\right )}{\sqrt {d x}}dx}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {1}{2} i \int -\frac {i e^{f x}}{\sqrt {d x}}dx-\frac {1}{2} i \int \frac {i e^{-f x}}{\sqrt {d x}}dx\right )}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {1}{2} \int \frac {e^{-f x}}{\sqrt {d x}}dx+\frac {1}{2} \int \frac {e^{f x}}{\sqrt {d x}}dx\right )}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\int e^{f x}d\sqrt {d x}}{d}\right )}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{2 f}\right )}{2 f}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {3 i d \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{2 f}\right )}{2 f}\) |
(((3*I)/2)*d*((I*Sqrt[d*x]*Cosh[f*x])/f - ((I/2)*d*((Sqrt[Pi]*Erf[(Sqrt[f] *Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d *x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f])))/f))/f + ((d*x)^(3/2)*Sinh[f*x])/f
3.1.63.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20
method | result | size |
meijerg | \(-\frac {2 i \left (d x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (10 f x +15\right ) {\mathrm e}^{-f x}}{80 \sqrt {\pi }\, f^{2}}-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (-10 f x +15\right ) {\mathrm e}^{f x}}{80 \sqrt {\pi }\, f^{2}}+\frac {3 \left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {5}{2}}}+\frac {3 \left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {5}{2}}}\right )}{x^{\frac {3}{2}} \left (i f \right )^{\frac {3}{2}} f}\) | \(133\) |
-2*I*(d*x)^(3/2)/x^(3/2)*2^(1/2)/(I*f)^(3/2)*Pi^(1/2)/f*(-1/80/Pi^(1/2)*x^ (1/2)*2^(1/2)*(I*f)^(5/2)*(10*f*x+15)/f^2*exp(-f*x)-1/80/Pi^(1/2)*x^(1/2)* 2^(1/2)*(I*f)^(5/2)*(-10*f*x+15)/f^2*exp(f*x)+3/32*(I*f)^(5/2)*2^(1/2)/f^( 5/2)*erf(x^(1/2)*f^(1/2))+3/32*(I*f)^(5/2)*2^(1/2)/f^(5/2)*erfi(x^(1/2)*f^ (1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (77) = 154\).
Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.72 \[ \int (d x)^{3/2} \cosh (f x) \, dx=\frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - 2 \, {\left (2 \, d f^{2} x - {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right )^{2} - 2 \, {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) - {\left (2 \, d f^{2} x - 3 \, d f\right )} \sinh \left (f x\right )^{2} + 3 \, d f\right )} \sqrt {d x}}{8 \, {\left (f^{3} \cosh \left (f x\right ) + f^{3} \sinh \left (f x\right )\right )}} \]
1/8*(3*sqrt(pi)*(d^2*cosh(f*x) + d^2*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sq rt(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))
Result contains complex when optimal does not.
Time = 11.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.18 \[ \int (d x)^{3/2} \cosh (f x) \, dx=\frac {5 d^{\frac {3}{2}} x^{\frac {3}{2}} \sinh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{4 f \Gamma \left (\frac {9}{4}\right )} - \frac {15 d^{\frac {3}{2}} \sqrt {x} \cosh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{8 f^{2} \Gamma \left (\frac {9}{4}\right )} + \frac {15 \sqrt {2} \sqrt {\pi } d^{\frac {3}{2}} e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{16 f^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
5*d**(3/2)*x**(3/2)*sinh(f*x)*gamma(5/4)/(4*f*gamma(9/4)) - 15*d**(3/2)*sq rt(x)*cosh(f*x)*gamma(5/4)/(8*f**2*gamma(9/4)) + 15*sqrt(2)*sqrt(pi)*d**(3 /2)*exp(-I*pi/4)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*ga mma(5/4)/(16*f**(5/2)*gamma(9/4))
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (77) = 154\).
Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.57 \[ \int (d x)^{3/2} \cosh (f x) \, dx=\frac {16 \, \left (d x\right )^{\frac {5}{2}} \cosh \left (f x\right ) + \frac {f {\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{3} \sqrt {\frac {f}{d}}} + \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{3} \sqrt {-\frac {f}{d}}} - \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} - 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} + 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (-f x\right )}}{f^{3}}\right )}}{d}}{40 \, d} \]
1/40*(16*(d*x)^(5/2)*cosh(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
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.31 \[ \int (d x)^{3/2} \cosh (f x) \, dx=-\frac {1}{8} \, d {\left (\frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f^{2}} + \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}}{d^{2}} + \frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f^{2}} - \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}}}{d^{2}}\right )} \]
-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)
Timed out. \[ \int (d x)^{3/2} \cosh (f x) \, dx=\int \mathrm {cosh}\left (f\,x\right )\,{\left (d\,x\right )}^{3/2} \,d x \]