Optimal. Leaf size=244 \[ \frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 x \sqrt {a x-1} \sqrt {a x+1}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.76, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205} \[ \frac {16 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {a x-1} \sqrt {a x+1}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5666
Rule 5668
Rule 5775
Rubi steps
\begin {align*} \int \frac {x^3}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac {6 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {64}{15} \int \frac {x^3}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac {8 \int \frac {x}{\cosh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac {128 \operatorname {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}-\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac {8 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}+\frac {64 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {64 \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {32 \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}-\frac {16 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac {16 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}+\frac {64 \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.59, size = 291, normalized size = 1.19 \[ \frac {e^{-4 \cosh ^{-1}(a x)} \left (-3 e^{8 \cosh ^{-1}(a x)}-64 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2+64 \cosh ^{-1}(a x)^2-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-8 \cosh ^{-1}(a x)+128 e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )-8 e^{2 \cosh ^{-1}(a x)} \left (4 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-4 \cosh ^{-1}(a x)^2+3 a x \sqrt {\frac {a x-1}{a x+1}} (a x+1) e^{2 \cosh ^{-1}(a x)}+e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)-4 \sqrt {2} e^{2 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )+4 \sqrt {2} e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )\right )-128 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )+3\right )}{120 a^4 \cosh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\mathrm {arccosh}\left (a x \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {acosh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________