Optimal. Leaf size=82 \[ -\frac {3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt {a x^2+a}}-\frac {(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac {(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5115, 5114} \[ -\frac {3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt {a x^2+a}}-\frac {(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac {(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5114
Rule 5115
Rubi steps
\begin {align*} \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx &=-\frac {e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}+\frac {10 \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{5/2}} \, dx}{13 a}\\ &=-\frac {e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac {e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}+\frac {6 \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx}{13 a^2}\\ &=-\frac {e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac {e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}-\frac {3 e^{\cot ^{-1}(x)} (1-x)}{13 a^3 \sqrt {a+a x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 95, normalized size = 1.16 \[ \frac {e^{\cot ^{-1}(x)} \left (-39 \sqrt {\frac {1}{x^2}+1} x \cos \left (3 \cot ^{-1}(x)\right )+5 \sqrt {\frac {1}{x^2}+1} x \cos \left (5 \cot ^{-1}(x)\right )+13 \sqrt {\frac {1}{x^2}+1} x \sin \left (3 \cot ^{-1}(x)\right )-\sqrt {\frac {1}{x^2}+1} x \sin \left (5 \cot ^{-1}(x)\right )+130 x-130\right )}{416 a^3 \sqrt {a \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 68, normalized size = 0.83 \[ \frac {{\left (6 \, x^{5} - 6 \, x^{4} + 18 \, x^{3} - 14 \, x^{2} + 17 \, x - 9\right )} \sqrt {a x^{2} + a} e^{\operatorname {arccot}\relax (x)}}{26 \, {\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 45, normalized size = 0.55 \[ \frac {\left (x^{2}+1\right ) \left (6 x^{5}-6 x^{4}+18 x^{3}-14 x^{2}+17 x -9\right ) {\mathrm e}^{\mathrm {arccot}\relax (x )}}{26 \left (a \,x^{2}+a \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.72, size = 58, normalized size = 0.71 \[ -\frac {9\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}-17\,x\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}+14\,x^2\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}-18\,x^3\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}+6\,x^4\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}-6\,x^5\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}}{26\,a\,{\left (a\,x^2+a\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________