Optimal. Leaf size=283 \[ \frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \text {ArcSin}(a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6294, 6264,
100, 155, 148, 41, 222} \begin {gather*} \frac {(a x+1)^2}{7 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \text {ArcSin}(a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 100
Rule 148
Rule 155
Rule 222
Rule 6264
Rule 6294
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x^7}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^7}{(1-a x)^{9/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^5 (6+8 a x)}{(1-a x)^{7/2} (1+a x)^{5/2}} \, dx}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^4 \left (-70 a-54 a^2 x\right )}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{35 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^3 \left (496 a^2+286 a^3 x\right )}{(1-a x)^{3/2} (1+a x)^{5/2}} \, dx}{105 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^2 \left (-2346 a^3-1068 a^4 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{105 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x \left (-2556 a^4-1926 a^5 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \sin ^{-1}(a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 133, normalized size = 0.47 \begin {gather*} \frac {-432+654 a x+636 a^2 x^2-1226 a^3 x^3-74 a^4 x^4+562 a^5 x^5-105 a^6 x^6-210 (-1+a x)^3 (1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{105 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^3 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(571\) vs.
\(2(253)=506\).
time = 0.86, size = 572, normalized size = 2.02
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a^{2} c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}-\frac {\left (\frac {2 \ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{7} \sqrt {a^{2} c}}-\frac {39 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{140 a^{11} c \left (x -\frac {1}{a}\right )^{3}}-\frac {1753 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{1680 a^{10} c \left (x -\frac {1}{a}\right )^{2}}-\frac {3061 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{840 a^{9} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{48 a^{10} c \left (x +\frac {1}{a}\right )^{2}}+\frac {7 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{24 a^{9} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{28 a^{12} c \left (x -\frac {1}{a}\right )^{4}}\right ) a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\) | \(384\) |
default | \(-\frac {\left (105 x^{7} c^{\frac {7}{2}} a^{7} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}+96 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} x^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}-96 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{5} x^{5}-392 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{5} x^{5}-240 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{4} x^{4}+1540 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{4} x^{4}+210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{6} c x +240 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{3} x^{3}+350 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{3} x^{3}-210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{5} c +180 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{2} x^{2}-1470 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{2} x^{2}-180 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a x -42 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a x -30 c^{\frac {7}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}}+462 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x +1\right )}{105 \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} a^{8} c^{\frac {7}{2}}}\) | \(572\) |
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 [A]
time = 0.47, size = 497, normalized size = 1.76 \begin {gather*} \left [\frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {210 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \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*} - \int \frac {a x}{a c^{3} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx - \int \frac {1}{a c^{3} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________