Optimal. Leaf size=329 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.45, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5904, 5912,
5942, 5907, 3393, 3388, 2211, 2235, 2236, 5953, 5556} \begin {gather*} -\frac {2 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5556
Rule 5904
Rule 5907
Rule 5912
Rule 5942
Rule 5953
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int \frac {(-1+a x)^{3/2} (1+a x)^{3/2}}{\cosh ^{-1}(a x)^{5/2}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {\left (8 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )}{\cosh ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{8 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 317, normalized size = 0.96 \begin {gather*} -\frac {c e^{-4 \cosh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \left (-1-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}+16 a^2 e^{4 \cosh ^{-1}(a x)} x^2+8 \cosh ^{-1}(a x)-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+64 a e^{4 \cosh ^{-1}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)+64 a^2 e^{4 \cosh ^{-1}(a x)} x^2 \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)-16 e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )-16 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )\right )}{24 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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