Optimal. Leaf size=143 \[ \frac{x^6 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (7 a x+6)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x^2 (35 a x+24)}{15 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{(35 a x+32) \sqrt{1-a^2 x^2}}{10 a^8 c^3}-\frac{7 \sin ^{-1}(a x)}{2 a^8 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18099, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 819, 780, 216} \[ \frac{x^6 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (7 a x+6)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x^2 (35 a x+24)}{15 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{(35 a x+32) \sqrt{1-a^2 x^2}}{10 a^8 c^3}-\frac{7 \sin ^{-1}(a x)}{2 a^8 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6148
Rule 819
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^7 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{x^5 (6+7 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac{x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{x^3 (24+35 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac{x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x^2 (24+35 a x)}{15 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{x (48+105 a x)}{\sqrt{1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac{x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x^2 (24+35 a x)}{15 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{(32+35 a x) \sqrt{1-a^2 x^2}}{10 a^8 c^3}-\frac{7 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^7 c^3}\\ &=\frac{x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x^2 (24+35 a x)}{15 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{(32+35 a x) \sqrt{1-a^2 x^2}}{10 a^8 c^3}-\frac{7 \sin ^{-1}(a x)}{2 a^8 c^3}\\ \end{align*}
Mathematica [A] time = 0.0766133, size = 116, normalized size = 0.81 \[ \frac{-15 a^6 x^6-15 a^5 x^5+176 a^4 x^4+4 a^3 x^3-249 a^2 x^2-105 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+9 a x+96}{30 a^8 c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.053, size = 283, normalized size = 2. \begin{align*}{\frac{x}{2\,{c}^{3}{a}^{7}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{7}{2\,{c}^{3}{a}^{7}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{c}^{3}{a}^{8}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{20\,{c}^{3}{a}^{11}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-{\frac{7}{15\,{c}^{3}{a}^{10}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{773}{240\,{c}^{3}{a}^{9}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{1}{24\,{c}^{3}{a}^{10} \left ( x+{a}^{-1} \right ) ^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{31}{48\,{c}^{3}{a}^{9} \left ( x+{a}^{-1} \right ) }\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a \int \frac{x^{8}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} + \frac{\frac{5 \, \sqrt{-a^{2} x^{2} + 1}}{c^{3}} + \frac{5 \, a^{2} x^{2} + 15 \,{\left (a^{2} x^{2} - 1\right )}^{2} - 4}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} c^{3}}}{5 \, a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69224, size = 485, normalized size = 3.39 \begin{align*} \frac{96 \, a^{5} x^{5} - 96 \, a^{4} x^{4} - 192 \, a^{3} x^{3} + 192 \, a^{2} x^{2} + 96 \, a x + 210 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (15 \, a^{6} x^{6} + 15 \, a^{5} x^{5} - 176 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 249 \, a^{2} x^{2} - 9 \, a x - 96\right )} \sqrt{-a^{2} x^{2} + 1} - 96}{30 \,{\left (a^{13} c^{3} x^{5} - a^{12} c^{3} x^{4} - 2 \, a^{11} c^{3} x^{3} + 2 \, a^{10} c^{3} x^{2} + a^{9} c^{3} x - a^{8} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{7}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{8}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )} x^{7}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]