Optimal. Leaf size=106 \[ -\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6265, 21, 100,
156, 162, 65, 214, 212} \begin {gather*} -\frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )-\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rule 214
Rule 6265
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx &=\int \frac {(1-a x) \sqrt {c-a c x}}{x^3 (1+a x)} \, dx\\ &=\frac {\int \frac {(c-a c x)^{3/2}}{x^3 (1+a x)} \, dx}{c}\\ &=-\frac {\sqrt {c-a c x}}{2 x^2}-\frac {\int \frac {\frac {9 a c^2}{2}-\frac {7}{2} a^2 c^2 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{2 c}\\ &=-\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {\int \frac {\frac {23 a^2 c^3}{4}-\frac {9}{4} a^3 c^3 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{2 c^2}\\ &=-\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {1}{8} \left (23 a^2 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx-\left (4 a^3 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {1}{4} (23 a) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )+\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=-\frac {\sqrt {c-a c x}}{2 x^2}+\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 93, normalized size = 0.88 \begin {gather*} \frac {(-2+9 a x) \sqrt {c-a c x}}{4 x^2}-\frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 94, normalized size = 0.89
method | result | size |
risch | \(-\frac {\left (9 a^{2} x^{2}-11 a x +2\right ) c}{4 x^{2} \sqrt {-c \left (a x -1\right )}}+\frac {a^{2} \left (-\frac {46 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {32 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}\right ) c}{8}\) | \(84\) |
derivativedivides | \(-2 c^{2} a^{2} \left (\frac {\frac {\frac {9 \left (-c x a +c \right )^{\frac {3}{2}}}{8}-\frac {7 c \sqrt {-c x a +c}}{8}}{c^{2} x^{2} a^{2}}+\frac {23 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )\) | \(94\) |
default | \(-2 c^{2} a^{2} \left (\frac {\frac {\frac {9 \left (-c x a +c \right )^{\frac {3}{2}}}{8}-\frac {7 c \sqrt {-c x a +c}}{8}}{c^{2} x^{2} a^{2}}+\frac {23 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 152, normalized size = 1.43 \begin {gather*} -\frac {1}{8} \, a^{2} c^{2} {\left (\frac {2 \, {\left (9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} - 7 \, \sqrt {-a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c + 2 \, {\left (a c x - c\right )} c^{2} + c^{3}} + \frac {16 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {3}{2}}} - \frac {23 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 204, normalized size = 1.92 \begin {gather*} \left [\frac {16 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 23 \, a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{8 \, x^{2}}, -\frac {16 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 23 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{4 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (99) = 198\).
time = 13.13, size = 352, normalized size = 3.32 \begin {gather*} - \frac {10 a^{2} c^{4} \sqrt {- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac {6 a^{2} c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} - \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} - \frac {3 a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {3 a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {8 a^{2} c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 \sqrt {2} a^{2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {3 a \sqrt {- a c x + c}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 106, normalized size = 1.00 \begin {gather*} -\frac {4 \, \sqrt {2} a^{2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \frac {23 \, a^{2} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c - 7 \, \sqrt {-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 88, normalized size = 0.83 \begin {gather*} \frac {7\,\sqrt {c-a\,c\,x}}{4\,x^2}+\frac {a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,23{}\mathrm {i}}{4}-\frac {9\,{\left (c-a\,c\,x\right )}^{3/2}}{4\,c\,x^2}-\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________