Optimal. Leaf size=476 \[ -\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.62, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 15, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used =
{5932, 5901, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5914, 39, 5936, 5916, 5569, 4267}
\begin {gather*} \frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \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^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 4267
Rule 5569
Rule 5899
Rule 5901
Rule 5912
Rule 5913
Rule 5914
Rule 5916
Rule 5932
Rule 5936
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \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 \left (-1+c^2 x^2\right )^2} \, dx}{d^2 \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}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \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 )}{d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \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 \left (-1+c^2 x^2\right )} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 c^2 \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^2 \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}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^3 \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^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{3 d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c \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^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{3 d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 b c \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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{3 d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 c \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^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{3 d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \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^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c \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^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{3 d^2 \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^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b c \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^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 2.02, size = 457, normalized size = 0.96 \begin {gather*} \frac {c \left (\frac {a^2 \left (3-12 c^2 x^2+8 c^4 x^4\right )}{c x \left (-1+c^2 x^2\right )}+a b \left (10 c x \cosh ^{-1}(c x)-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+2 c x \cosh ^{-1}(c x)}{-1+c^2 x^2}-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\frac {3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{c x}+3 \log (c x)+5 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )+b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {\cosh ^{-1}(c x)}{1-c^2 x^2}-8 \cosh ^{-1}(c x)^2-\frac {c x \cosh ^{-1}(c x)^2}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+\frac {5 c x \cosh ^{-1}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)^2}{c x}-10 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-6 \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+5 \text {PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )\right )\right )}{3 d^2 \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. \(3793\) vs.
\(2(470)=940\).
time = 2.02, size = 3794, normalized size = 7.97
method | result | size |
default | \(\text {Expression too large to display}\) | \(3794\) |
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^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{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^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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