Optimal. Leaf size=111 \[ \frac {3 \sqrt {\pi } d^{3/2} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 \sqrt {\pi } d^{3/2} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}-\frac {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {(d x)^{3/2} \sinh (f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, 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 = {3296, 3307, 2180, 2204, 2205} \[ \frac {3 \sqrt {\pi } d^{3/2} \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}+\frac {3 \sqrt {\pi } d^{3/2} \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{8 f^{5/2}}-\frac {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {(d x)^{3/2} \sinh (f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3296
Rule 3307
Rubi steps
\begin {align*} \int (d x)^{3/2} \cosh (f x) \, dx &=\frac {(d x)^{3/2} \sinh (f x)}{f}-\frac {(3 d) \int \sqrt {d x} \sinh (f x) \, dx}{2 f}\\ &=-\frac {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {\left (3 d^2\right ) \int \frac {\cosh (f x)}{\sqrt {d x}} \, dx}{4 f^2}\\ &=-\frac {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {(d x)^{3/2} \sinh (f x)}{f}+\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 {3 d \sqrt {d x} \cosh (f x)}{2 f^2}+\frac {(d x)^{3/2} \sinh (f x)}{f}+\frac {(3 d) \operatorname {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{4 f^2}+\frac {(3 d) \operatorname {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{4 f^2}\\ &=-\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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 51, normalized size = 0.46 \[ \frac {d^2 \left (\sqrt {-f x} \Gamma \left (\frac {5}{2},-f x\right )-\sqrt {f x} \Gamma \left (\frac {5}{2},f x\right )\right )}{2 f^3 \sqrt {d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 191, normalized size = 1.72 \[ \frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (f x\right ) + d^{2} \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - 2 \, {\left (2 \, d f^{2} x - {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right )^{2} - 2 \, {\left (2 \, d f^{2} x - 3 \, d f\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) - {\left (2 \, d f^{2} x - 3 \, d f\right )} \sinh \left (f x\right )^{2} + 3 \, d f\right )} \sqrt {d x}}{8 \, {\left (f^{3} \cosh \left (f x\right ) + f^{3} \sinh \left (f x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 145, normalized size = 1.31 \[ -\frac {1}{8} \, d {\left (\frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f^{2}} + \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}}{d^{2}} + \frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f^{2}} - \frac {2 \, {\left (2 \, \sqrt {d x} d^{2} f x - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}}}{d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.09, size = 133, normalized size = 1.20 \[ -\frac {2 i \left (d x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (10 f x +15\right ) {\mathrm e}^{-f x}}{80 \sqrt {\pi }\, f^{2}}-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (-10 f x +15\right ) {\mathrm e}^{f x}}{80 \sqrt {\pi }\, f^{2}}+\frac {3 \left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {5}{2}}}+\frac {3 \left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{32 f^{\frac {5}{2}}}\right )}{x^{\frac {3}{2}} \left (i f \right )^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.32, size = 174, normalized size = 1.57 \[ \frac {16 \, \left (d x\right )^{\frac {5}{2}} \cosh \left (f x\right ) + \frac {f {\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{3} \sqrt {\frac {f}{d}}} + \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{3} \sqrt {-\frac {f}{d}}} - \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} - 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {2 \, {\left (4 \, \left (d x\right )^{\frac {5}{2}} d f^{2} + 10 \, \left (d x\right )^{\frac {3}{2}} d^{2} f + 15 \, \sqrt {d x} d^{3}\right )} e^{\left (-f x\right )}}{f^{3}}\right )}}{d}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cosh}\left (f\,x\right )\,{\left (d\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 18.03, size = 131, normalized size = 1.18 \[ \frac {5 d^{\frac {3}{2}} x^{\frac {3}{2}} \sinh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{4 f \Gamma \left (\frac {9}{4}\right )} - \frac {15 d^{\frac {3}{2}} \sqrt {x} \cosh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{8 f^{2} \Gamma \left (\frac {9}{4}\right )} + \frac {15 \sqrt {2} \sqrt {\pi } d^{\frac {3}{2}} e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{16 f^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________