Optimal. Leaf size=58 \[ \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5904, 5953,
5556, 3382} \begin {gather*} \frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac {c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3382
Rule 5556
Rule 5904
Rule 5953
Rubi steps
\begin {align*} \int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-(3 a c) \int \frac {x \sqrt {-1+a x} \sqrt {1+a x}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(58)=116\).
time = 1.29, size = 217, normalized size = 3.74 \begin {gather*} \frac {c \sqrt {-1+a x} \left (4 \left (\frac {-1+a x}{1+a x}\right )^{5/2} (1+a x)^5-3 \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )-(-1+a x) \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right ) \left (1+a x-4 \sqrt {-1+a x} \sqrt {1+a x} \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )+4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+(1-a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right ) \log \left (\cosh ^{-1}(a x)\right )\right )}{4 a \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{5/2} \cosh ^{-1}(a x)} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.12, size = 61, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {c \left (3 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a \,\mathrm {arccosh}\left (a x \right )}\) | \(61\) |
default | \(\frac {c \left (3 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a \,\mathrm {arccosh}\left (a x \right )}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {c-a^2\,c\,x^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________