Optimal. Leaf size=204 \[ -\frac {a \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 (a-b)^{9/2} f}-\frac {\left (5 a^2+10 a b-b^2\right ) \cos (e+f x)}{5 (a-b)^4 f}+\frac {(10 a-3 b) \cos ^3(e+f x)}{15 (a-b)^3 f}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.22, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3745, 473, 467,
1275, 211} \begin {gather*} -\frac {\left (5 a^2+10 a b-b^2\right ) \cos (e+f x)}{5 f (a-b)^4}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 f (a-b)^4 \left (a+b \sec ^2(e+f x)-b\right )}-\frac {a \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 f (a-b)^{9/2}}+\frac {(10 a-3 b) \cos ^3(e+f x)}{15 f (a-b)^3}-\frac {\cos ^5(e+f x)}{5 f (a-b) \left (a+b \sec ^2(e+f x)-b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 467
Rule 473
Rule 1275
Rule 3745
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {-10 a+3 b+5 (a-b) x^2}{x^4 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{5 (a-b) f}\\ &=-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \text {Subst}\left (\int \frac {\frac {2 (10 a-3 b)}{(a-b) b}-\frac {2 \left (5 a^2+2 b^2\right ) x^2}{(a-b)^2 b}+\frac {\left (5 a^2+2 b^2\right ) x^4}{(a-b)^3}}{x^4 \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{10 (a-b) f}\\ &=-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \text {Subst}\left (\int \left (\frac {2 (10 a-3 b)}{(a-b)^2 b x^4}-\frac {2 \left (5 a^2+10 a b-b^2\right )}{(a-b)^3 b x^2}+\frac {5 a (3 a+4 b)}{(a-b)^3 \left (a-b+b x^2\right )}\right ) \, dx,x,\sec (e+f x)\right )}{10 (a-b) f}\\ &=-\frac {\left (5 a^2+10 a b-b^2\right ) \cos (e+f x)}{5 (a-b)^4 f}+\frac {(10 a-3 b) \cos ^3(e+f x)}{15 (a-b)^3 f}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {(a b (3 a+4 b)) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{2 (a-b)^4 f}\\ &=-\frac {a \sqrt {b} (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 (a-b)^{9/2} f}-\frac {\left (5 a^2+10 a b-b^2\right ) \cos (e+f x)}{5 (a-b)^4 f}+\frac {(10 a-3 b) \cos ^3(e+f x)}{15 (a-b)^3 f}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \left (5 a^2+2 b^2\right ) \sec (e+f x)}{10 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 2.51, size = 215, normalized size = 1.05 \begin {gather*} \frac {\frac {120 a \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {a-b}-\sqrt {a} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )}{(a-b)^{9/2}}+\frac {120 a \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {a-b}+\sqrt {a} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )}{(a-b)^{9/2}}+\frac {-30 \cos (e+f x) \left (18 a b+b^2+a^2 \left (5+\frac {8 b}{a+b+(a-b) \cos (2 (e+f x))}\right )\right )+(a-b) (5 (5 a+3 b) \cos (3 (e+f x))+3 (-a+b) \cos (5 (e+f x)))}{(a-b)^4}}{240 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 197, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {a^{2} \left (\cos ^{5}\left (f x +e \right )\right )}{5}-\frac {2 a b \left (\cos ^{5}\left (f x +e \right )\right )}{5}+\frac {b^{2} \left (\cos ^{5}\left (f x +e \right )\right )}{5}-\frac {2 a^{2} \left (\cos ^{3}\left (f x +e \right )\right )}{3}+\frac {2 a b \left (\cos ^{3}\left (f x +e \right )\right )}{3}+a^{2} \cos \left (f x +e \right )+2 a b \cos \left (f x +e \right )}{\left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2}}+\frac {a b \left (-\frac {a \cos \left (f x +e \right )}{2 \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}+\frac {\left (3 a +4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{\left (a -b \right )^{4}}}{f}\) | \(197\) |
default | \(\frac {-\frac {\frac {a^{2} \left (\cos ^{5}\left (f x +e \right )\right )}{5}-\frac {2 a b \left (\cos ^{5}\left (f x +e \right )\right )}{5}+\frac {b^{2} \left (\cos ^{5}\left (f x +e \right )\right )}{5}-\frac {2 a^{2} \left (\cos ^{3}\left (f x +e \right )\right )}{3}+\frac {2 a b \left (\cos ^{3}\left (f x +e \right )\right )}{3}+a^{2} \cos \left (f x +e \right )+2 a b \cos \left (f x +e \right )}{\left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2}}+\frac {a b \left (-\frac {a \cos \left (f x +e \right )}{2 \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}+\frac {\left (3 a +4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{\left (a -b \right )^{4}}}{f}\) | \(197\) |
risch | \(-\frac {5 \,{\mathrm e}^{3 i \left (f x +e \right )} a}{96 \left (-a +b \right )^{3} f}-\frac {{\mathrm e}^{3 i \left (f x +e \right )} b}{32 \left (-a +b \right )^{3} f}-\frac {5 \,{\mathrm e}^{i \left (f x +e \right )} a^{2}}{16 f \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2}}-\frac {9 \,{\mathrm e}^{i \left (f x +e \right )} a b}{8 f \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2}}-\frac {{\mathrm e}^{i \left (f x +e \right )} b^{2}}{16 f \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2}}-\frac {5 \,{\mathrm e}^{-i \left (f x +e \right )} a^{2}}{16 \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right ) f}-\frac {9 \,{\mathrm e}^{-i \left (f x +e \right )} a b}{8 \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right ) f}-\frac {{\mathrm e}^{-i \left (f x +e \right )} b^{2}}{16 \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right ) f}-\frac {5 \,{\mathrm e}^{-3 i \left (f x +e \right )} a}{96 \left (-a^{3}+3 a^{2} b -3 a \,b^{2}+b^{3}\right ) f}-\frac {{\mathrm e}^{-3 i \left (f x +e \right )} b}{32 \left (-a^{3}+3 a^{2} b -3 a \,b^{2}+b^{3}\right ) f}-\frac {b \,a^{2} \left ({\mathrm e}^{3 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{\left (a^{2}-2 a b +b^{2}\right )^{2} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}-\frac {3 i \sqrt {b \left (a -b \right )}\, a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 \left (a -b \right )^{5} f}-\frac {i \sqrt {b \left (a -b \right )}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{\left (a -b \right )^{5} f}+\frac {3 i \sqrt {b \left (a -b \right )}\, a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 \left (a -b \right )^{5} f}+\frac {i \sqrt {b \left (a -b \right )}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{\left (a -b \right )^{5} f}-\frac {\cos \left (5 f x +5 e \right )}{80 f \left (a -b \right )^{2}}\) | \(743\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.25, size = 609, normalized size = 2.99 \begin {gather*} \left [-\frac {12 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{7} - 4 \, {\left (10 \, a^{3} - 23 \, a^{2} b + 16 \, a b^{2} - 3 \, b^{3}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (3 \, a^{2} b + 4 \, a b^{2} + {\left (3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a - b}} \log \left (\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a - b\right )} \sqrt {-\frac {b}{a - b}} \cos \left (f x + e\right ) - b}{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}\right ) + 30 \, {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )}{60 \, {\left ({\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )} f\right )}}, -\frac {6 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (10 \, a^{3} - 23 \, a^{2} b + 16 \, a b^{2} - 3 \, b^{3}\right )} \cos \left (f x + e\right )^{5} + 10 \, {\left (3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (3 \, a^{2} b + 4 \, a b^{2} + {\left (3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a - b}} \arctan \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a - b}} \cos \left (f x + e\right )}{b}\right ) + 15 \, {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )}{30 \, {\left ({\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 561 vs.
\(2 (193) = 386\).
time = 0.85, size = 561, normalized size = 2.75 \begin {gather*} -\frac {\frac {15 \, {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b - b^{2}}} + \frac {30 \, {\left (a^{2} b + \frac {a^{2} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )}}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}} - \frac {4 \, {\left (8 \, a^{2} + 34 \, a b + 3 \, b^{2} - \frac {40 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {140 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {80 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {160 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {180 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {30 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {15 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 1\right )}^{5}}}{30 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.49, size = 1049, normalized size = 5.14 \begin {gather*} -\frac {\frac {16\,a^3+83\,a^2\,b+6\,a\,b^2}{15\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (32\,a^3-83\,a^2\,b+366\,a\,b^2\right )}{3\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (16\,a^3+223\,a^2\,b+1336\,a\,b^2\right )}{15\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (6\,a^2\,b+11\,a\,b^2+4\,b^3\right )}{\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (-12\,a^3+32\,a^2\,b+73\,a\,b^2+12\,b^3\right )}{3\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (24\,a^3+134\,a^2\,b+145\,a\,b^2+12\,b^3\right )}{15\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (4\,b^2+3\,a\,b\right )}{\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+\left (3\,a+4\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+\left (a+20\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+\left (40\,b-5\,a\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+\left (40\,b-5\,a\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+\left (a+20\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+\left (3\,a+4\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\right )}-\frac {a\,\sqrt {b}\,\mathrm {atan}\left (\frac {{\left (a-b\right )}^9\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {\sqrt {b}\,\left (3\,a+4\,b\right )\,\left (24\,a^{12}\,b-160\,a^{11}\,b^2+416\,a^{10}\,b^3-448\,a^9\,b^4-112\,a^8\,b^5+896\,a^7\,b^6-1120\,a^6\,b^7+704\,a^5\,b^8-232\,a^4\,b^9+32\,a^3\,b^{10}\right )}{4\,{\left (a-b\right )}^{17/2}}-\frac {a\,\sqrt {b}\,\left (a-2\,b\right )\,{\left (3\,a+4\,b\right )}^2\,\left (-16\,a^{15}+224\,a^{14}\,b-1440\,a^{13}\,b^2+5632\,a^{12}\,b^3-14960\,a^{11}\,b^4+28512\,a^{10}\,b^5-40128\,a^9\,b^6+42240\,a^8\,b^7-33264\,a^7\,b^8+19360\,a^6\,b^9-8096\,a^5\,b^{10}+2304\,a^4\,b^{11}-400\,a^3\,b^{12}+32\,a^2\,b^{13}\right )}{32\,{\left (a-b\right )}^{27/2}}\right )-\frac {a\,\sqrt {b}\,\left (a-2\,b\right )\,{\left (3\,a+4\,b\right )}^2\,\left (16\,a^{15}-192\,a^{14}\,b+1056\,a^{13}\,b^2-3520\,a^{12}\,b^3+7920\,a^{11}\,b^4-12672\,a^{10}\,b^5+14784\,a^9\,b^6-12672\,a^8\,b^7+7920\,a^7\,b^8-3520\,a^6\,b^9+1056\,a^5\,b^{10}-192\,a^4\,b^{11}+16\,a^3\,b^{12}\right )}{32\,{\left (a-b\right )}^{27/2}}\right )}{9\,a^{14}\,b-48\,a^{13}\,b^2+76\,a^{12}\,b^3+40\,a^{11}\,b^4-266\,a^{10}\,b^5+280\,a^9\,b^6+28\,a^8\,b^7-296\,a^7\,b^8+265\,a^6\,b^9-104\,a^5\,b^{10}+16\,a^4\,b^{11}}\right )\,\left (3\,a+4\,b\right )}{2\,f\,{\left (a-b\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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