Optimal. Leaf size=479 \[ -\frac {\sqrt {\pi } d^2 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 \sqrt {\frac {\pi }{2}} d^2 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {\sqrt {\frac {\pi }{2}} d^2 e^{\frac {8 a}{b}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\pi } d^2 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 \sqrt {\frac {\pi }{2}} d^2 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {\sqrt {\frac {\pi }{2}} d^2 e^{-\frac {8 a}{b}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 2.13, antiderivative size = 491, normalized size of antiderivative = 1.03, number of steps used = 32, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5776, 5781, 5448, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } d^2 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 \sqrt {\frac {\pi }{2}} d^2 e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {\sqrt {\frac {\pi }{2}} d^2 e^{\frac {8 a}{b}} \text {Erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{\frac {6 a}{b}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\pi } d^2 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 \sqrt {\frac {\pi }{2}} d^2 e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {\sqrt {\frac {\pi }{2}} d^2 e^{-\frac {8 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{-\frac {6 a}{b}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {2 d^2 x^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5776
Rule 5781
Rubi steps
\begin {align*} \int \frac {x^3 \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^3 \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 (6 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 c}+\frac {\left (16 c d^2\right ) \int \frac {x^4 (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 x^3 \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 (6 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}+\frac {\left (16 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {2 d^2 x^3 \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 (6 d^2\right ) \operatorname {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^4}+\frac {\left (16 d^2\right ) \operatorname {Subst}\left (\int \left (\frac {3}{128 \sqrt {a+b x}}-\frac {\cosh (4 x)}{32 \sqrt {a+b x}}+\frac {\cosh (8 x)}{128 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {2 d^2 x^3 \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 \operatorname {Subst}\left (\int \frac {\cosh (8 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}-\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (6 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^4}\\ &=-\frac {2 d^2 x^3 \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 \operatorname {Subst}\left (\int \frac {e^{-8 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{8 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}-\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^4}-\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4}\\ &=-\frac {2 d^2 x^3 \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 \operatorname {Subst}\left (\int e^{\frac {8 a}{b}-\frac {8 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}+\frac {d^2 \operatorname {Subst}\left (\int e^{-\frac {8 a}{b}+\frac {8 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}-\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {6 a}{b}-\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 b^2 c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 b^2 c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 b^2 c^4}-\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {6 a}{b}+\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 b^2 c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}-\frac {d^2 \operatorname {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^4}-\frac {d^2 \operatorname {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^4}\\ &=-\frac {2 d^2 x^3 \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 )}{32 b^{3/2} c^4}+\frac {3 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 )}{32 b^{3/2} c^4}+\frac {d^2 e^{\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\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 )}{32 b^{3/2} c^4}-\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 )}{32 b^{3/2} c^4}+\frac {3 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 )}{32 b^{3/2} c^4}+\frac {d^2 e^{-\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\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 )}{32 b^{3/2} c^4}\\ \end {align*}
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Mathematica [A] time = 3.78, size = 527, normalized size = 1.10 \[ -\frac {d^2 e^{-\frac {8 a}{b}} \left (128 c^4 x^4 e^{\frac {8 a}{b}} \sqrt {\frac {c x-1}{c x+1}}+128 c^3 x^3 e^{\frac {8 a}{b}} \sqrt {\frac {c x-1}{c x+1}}-26 e^{\frac {8 a}{b}} \sinh \left (2 \cosh ^{-1}(c x)\right )-18 e^{\frac {8 a}{b}} \sinh \left (4 \cosh ^{-1}(c x)\right )-2 e^{\frac {8 a}{b}} \sinh \left (6 \cosh ^{-1}(c x)\right )+e^{\frac {8 a}{b}} \sinh \left (8 \cosh ^{-1}(c x)\right )-\sqrt {2} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt {6} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-3 \sqrt {2} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt {2} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-2 e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\sqrt {6} e^{\frac {14 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt {2} e^{\frac {16 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{64 b c^4 \sqrt {a+b \cosh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ d^{2} \left (\int \frac {x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{5}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{7}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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