Optimal. Leaf size=363 \[ -\frac {2 d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2} \]
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Rubi [A]
time = 1.12, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps
used = 32, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5942, 5907,
3393, 3388, 2211, 2236, 2235, 5953, 5556} \begin {gather*} -\frac {\sqrt {\pi } d^2 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 \sqrt {\frac {\pi }{2}} d^2 e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{\frac {6 a}{b}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {\sqrt {\pi } d^2 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 \sqrt {\frac {\pi }{2}} d^2 e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{-\frac {6 a}{b}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5556
Rule 5907
Rule 5942
Rule 5953
Rubi steps
\begin {align*} \int \frac {x \left (d-c^2 d x^2\right )^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (2 d^2\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b c}+\frac {\left (12 c d^2\right ) \int \frac {x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}+\frac {\left (12 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {a+b x}}-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}+\frac {\cosh (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}+\frac {\left (12 d^2\right ) \text {Subst}\left (\int \left (\frac {1}{16 \sqrt {a+b x}}-\frac {\cosh (2 x)}{32 \sqrt {a+b x}}-\frac {\cosh (4 x)}{16 \sqrt {a+b x}}+\frac {\cosh (6 x)}{32 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d^2 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (6 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d^2 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}-\frac {d^2 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d^2 \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^2}-\frac {d^2 \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {6 a}{b}-\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {6 a}{b}+\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {d^2 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 d^2 x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}\\ \end {align*}
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Mathematica [A]
time = 4.80, size = 508, normalized size = 1.40 \begin {gather*} \frac {d^2 e^{-\frac {6 a}{b}} \left (16 e^{\frac {8 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+16 e^{\frac {4 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+\frac {\sqrt {b} \left (128 c^3 e^{\frac {6 a}{b}} x^3 \sqrt {\frac {-1+c x}{1+c x}}+128 c^4 e^{\frac {6 a}{b}} x^4 \sqrt {\frac {-1+c x}{1+c x}}+\sqrt {6} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-8 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-11 \sqrt {2} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+11 \sqrt {2} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+8 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\sqrt {6} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-42 e^{\frac {6 a}{b}} \sinh \left (2 \cosh ^{-1}(c x)\right )-8 e^{\frac {6 a}{b}} \sinh \left (4 \cosh ^{-1}(c x)\right )-2 e^{\frac {6 a}{b}} \sinh \left (6 \cosh ^{-1}(c x)\right )\right )}{\sqrt {a+b \cosh ^{-1}(c x)}}\right )}{32 b^{3/2} c^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
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*} d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {x\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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