Optimal. Leaf size=351 \[ \frac {5 \sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 0.92, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5695, 5781, 5448, 3307, 2180, 2204, 2205} \[ \frac {5 \sqrt {\pi } d^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5695
Rule 5781
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (10 c d^2\right ) \int \frac {x (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (10 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (10 d^2\right ) \operatorname {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{16 \sqrt {a+b x}}+\frac {\cosh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}\\ &=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}\\ \end {align*}
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Mathematica [A] time = 1.90, size = 387, normalized size = 1.10 \[ -\frac {d^2 e^{-\frac {5 a}{b}} \left (20 c x e^{\frac {5 a}{b}} \sqrt {\frac {c x-1}{c x+1}}+20 e^{\frac {5 a}{b}} \sqrt {\frac {c x-1}{c x+1}}-10 e^{\frac {5 a}{b}} \sinh \left (3 \cosh ^{-1}(c x)\right )+2 e^{\frac {5 a}{b}} \sinh \left (5 \cosh ^{-1}(c x)\right )+10 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )-\sqrt {5} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+5 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-10 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )-5 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{16 b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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