Optimal. Leaf size=328 \[ \frac {b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x}-\frac {4 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.33, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {5932, 5917,
5882, 3799, 2221, 2317, 2438, 5883, 97} \begin {gather*} -\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {4 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 (1-c x) (c x+1)}{3 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \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 3799
Rule 5882
Rule 5883
Rule 5917
Rule 5932
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \sqrt {d-c^2 d x^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 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 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} \, dx}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \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 x^2 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 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 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 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 x^2 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \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) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 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 x^2 \sqrt {d-c^2 d x^2}}+\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \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 \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 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 x^2 \sqrt {d-c^2 d x^2}}+\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt {d-c^2 d x^2}}-\frac {4 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 \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 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 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 x^2 \sqrt {d-c^2 d x^2}}+\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt {d-c^2 d x^2}}-\frac {4 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 \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 \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 (1-c x) (1+c x)}{3 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 x^2 \sqrt {d-c^2 d x^2}}+\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt {d-c^2 d x^2}}-\frac {4 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 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 370, normalized size = 1.13 \begin {gather*} \frac {-a^2-a^2 c^2 x^2+b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {-1+c x} \sqrt {1+c x}-b^2 (1+c x) \left (1-c x+2 c^2 x^2+2 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \cosh ^{-1}(c x)^2+b (1+c x) \cosh ^{-1}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}+2 a \left (-1+c x-2 c^2 x^2+2 c^3 x^3\right )-4 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (-1+\sqrt {1+c x}\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1+\sqrt {1+c x}\right )+2 b^2 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 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. \(2196\) vs.
\(2(310)=620\).
time = 3.76, size = 2197, normalized size = 6.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(2197\) |
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} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, 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\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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