Optimal. Leaf size=496 \[ \frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.76, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5932, 5899,
5913, 3797, 2221, 2317, 2438, 5912, 5916, 5569, 4267, 97} \begin {gather*} -\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 (1-c x) (c x+1)}{3 d x \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 97
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 4267
Rule 5569
Rule 5899
Rule 5912
Rule 5913
Rule 5916
Rule 5932
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {8 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {16 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.70, size = 529, normalized size = 1.07 \begin {gather*} \frac {a^2 \left (-1-4 c^2 x^2+8 c^4 x^4\right )+a b \left (6 c^4 x^4 \cosh ^{-1}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (c x+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)\right )+2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (5 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-c x \left (5 \log (c x)+3 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )\right )+b^2 \left (c^2 x^2-c^4 x^4+c x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)+3 c^4 x^4 \cosh ^{-1}(c x)^2+(-1+c x) (1+c x) \cosh ^{-1}(c x)^2+5 c^2 x^2 (-1+c x) (1+c x) \cosh ^{-1}(c x)^2-8 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)^2-6 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-10 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+5 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+3 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )\right )}{3 d x^3 \sqrt {d-c^2 d x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2866\) vs.
\(2(486)=972\).
time = 3.64, size = 2867, normalized size = 5.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(2867\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, 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 (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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