Optimal. Leaf size=180 \[ -\frac {5 \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 (a-b)^{9/2} f}-\frac {(a+2 b) \cos (e+f x)}{(a-b)^4 f}+\frac {\cos ^3(e+f x)}{3 (a-b)^3 f}-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b (7 a+4 b) \sec (e+f x)}{8 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.18, antiderivative size = 180, 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, 467,
1273, 1275, 211} \begin {gather*} -\frac {5 \sqrt {b} (3 a+4 b) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 f (a-b)^{9/2}}+\frac {\cos ^3(e+f x)}{3 f (a-b)^3}-\frac {(a+2 b) \cos (e+f x)}{f (a-b)^4}-\frac {b (7 a+4 b) \sec (e+f x)}{8 f (a-b)^4 \left (a+b \sec ^2(e+f x)-b\right )}-\frac {a b \sec (e+f x)}{4 f (a-b)^3 \left (a+b \sec ^2(e+f x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 467
Rule 1273
Rule 1275
Rule 3745
Rubi steps
\begin {align*} \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b \text {Subst}\left (\int \frac {\frac {4}{(a-b) b}-\frac {4 a x^2}{(a-b)^2 b}+\frac {3 a x^4}{(a-b)^3}}{x^4 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b (7 a+4 b) \sec (e+f x)}{8 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {8 (a-b) b-8 b (a+b) x^2+\frac {b^2 (7 a+4 b) x^4}{a-b}}{x^4 \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{8 (a-b)^3 b f}\\ &=-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b (7 a+4 b) \sec (e+f x)}{8 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \left (\frac {8 b}{x^4}-\frac {8 b (a+2 b)}{(a-b) x^2}+\frac {5 b^2 (3 a+4 b)}{(a-b) \left (a-b+b x^2\right )}\right ) \, dx,x,\sec (e+f x)\right )}{8 (a-b)^3 b f}\\ &=-\frac {(a+2 b) \cos (e+f x)}{(a-b)^4 f}+\frac {\cos ^3(e+f x)}{3 (a-b)^3 f}-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b (7 a+4 b) \sec (e+f x)}{8 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {(5 b (3 a+4 b)) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{8 (a-b)^4 f}\\ &=-\frac {5 \sqrt {b} (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 (a-b)^{9/2} f}-\frac {(a+2 b) \cos (e+f x)}{(a-b)^4 f}+\frac {\cos ^3(e+f x)}{3 (a-b)^3 f}-\frac {a b \sec (e+f x)}{4 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {b (7 a+4 b) \sec (e+f x)}{8 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 3.71, size = 230, normalized size = 1.28 \begin {gather*} \frac {\frac {15 \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 {15 \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 {2 \left (3 \cos (e+f x) \left (a \left (-3+\frac {4 b^2}{(a+b+(a-b) \cos (2 (e+f x)))^2}-\frac {9 b}{a+b+(a-b) \cos (2 (e+f x))}\right )+b \left (-9-\frac {4 b}{a+b+(a-b) \cos (2 (e+f x))}\right )\right )+(a-b) \cos (3 (e+f x))\right )}{(a-b)^4}}{24 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 196, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \left (\cos ^{3}\left (f x +e \right )\right )}{3}-\frac {b \left (\cos ^{3}\left (f x +e \right )\right )}{3}-\cos \left (f x +e \right ) a -2 b \cos \left (f x +e \right )}{\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}+\frac {5}{8} a b +\frac {1}{2} b^{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right )+\left (-\frac {7}{8} a b -\frac {1}{2} b^{2}\right ) \cos \left (f x +e \right )}{\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2}}+\frac {5 \left (3 a +4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{\left (a -b \right )^{4}}}{f}\) | \(196\) |
default | \(\frac {\frac {\frac {a \left (\cos ^{3}\left (f x +e \right )\right )}{3}-\frac {b \left (\cos ^{3}\left (f x +e \right )\right )}{3}-\cos \left (f x +e \right ) a -2 b \cos \left (f x +e \right )}{\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}+\frac {5}{8} a b +\frac {1}{2} b^{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right )+\left (-\frac {7}{8} a b -\frac {1}{2} b^{2}\right ) \cos \left (f x +e \right )}{\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2}}+\frac {5 \left (3 a +4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{\left (a -b \right )^{4}}}{f}\) | \(196\) |
risch | \(\frac {{\mathrm e}^{3 i \left (f x +e \right )}}{24 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )} a}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right ) f}-\frac {9 \,{\mathrm e}^{i \left (f x +e \right )} b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right ) f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )} a}{8 \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 )} 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}^{-3 i \left (f x +e \right )}}{24 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) f}+\frac {b \left (-9 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+5 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+4 b^{2} {\mathrm e}^{7 i \left (f x +e \right )}-27 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}-13 a b \,{\mathrm e}^{5 i \left (f x +e \right )}-4 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}-27 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}-13 a b \,{\mathrm e}^{3 i \left (f x +e \right )}-4 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}-9 a^{2} {\mathrm e}^{i \left (f x +e \right )}+5 a b \,{\mathrm e}^{i \left (f x +e \right )}+4 b^{2} {\mathrm e}^{i \left (f x +e \right )}\right )}{4 \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 )^{2} \left (-a^{3}+3 a^{2} b -3 a \,b^{2}+b^{3}\right ) \left (-a +b \right ) f}+\frac {15 i \sqrt {b \left (a -b \right )}\, \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 ) a}{16 \left (a -b \right )^{5} f}+\frac {5 i \sqrt {b \left (a -b \right )}\, \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}{4 \left (a -b \right )^{5} f}-\frac {15 i \sqrt {b \left (a -b \right )}\, \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 ) a}{16 \left (a -b \right )^{5} f}-\frac {5 i \sqrt {b \left (a -b \right )}\, \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}{4 \left (a -b \right )^{5} f}\) | \(761\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 378 vs.
\(2 (171) = 342\).
time = 3.75, size = 795, normalized size = 4.42 \begin {gather*} \left [\frac {16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{7} - 16 \, {\left (3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 50 \, {\left (3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 4 \, b^{3} + 2 \, {\left (3 \, a^{2} b + a b^{2} - 4 \, b^{3}\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 b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )}{48 \, {\left ({\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} f\right )}}, \frac {8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{7} - 8 \, {\left (3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 25 \, {\left (3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 4 \, b^{3} + 2 \, {\left (3 \, a^{2} b + a b^{2} - 4 \, b^{3}\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 b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )}{24 \, {\left ({\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\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. 563 vs.
\(2 (171) = 342\).
time = 1.15, size = 563, normalized size = 3.13 \begin {gather*} \frac {a^{6} f^{17} \cos \left (f x + e\right )^{3} - 6 \, a^{5} b f^{17} \cos \left (f x + e\right )^{3} + 15 \, a^{4} b^{2} f^{17} \cos \left (f x + e\right )^{3} - 20 \, a^{3} b^{3} f^{17} \cos \left (f x + e\right )^{3} + 15 \, a^{2} b^{4} f^{17} \cos \left (f x + e\right )^{3} - 6 \, a b^{5} f^{17} \cos \left (f x + e\right )^{3} + b^{6} f^{17} \cos \left (f x + e\right )^{3} - 3 \, a^{6} f^{17} \cos \left (f x + e\right ) + 9 \, a^{5} b f^{17} \cos \left (f x + e\right ) - 30 \, a^{3} b^{3} f^{17} \cos \left (f x + e\right ) + 45 \, a^{2} b^{4} f^{17} \cos \left (f x + e\right ) - 27 \, a b^{5} f^{17} \cos \left (f x + e\right ) + 6 \, b^{6} f^{17} \cos \left (f x + e\right )}{3 \, {\left (a^{9} f^{18} - 9 \, a^{8} b f^{18} + 36 \, a^{7} b^{2} f^{18} - 84 \, a^{6} b^{3} f^{18} + 126 \, a^{5} b^{4} f^{18} - 126 \, a^{4} b^{5} f^{18} + 84 \, a^{3} b^{6} f^{18} - 36 \, a^{2} b^{7} f^{18} + 9 \, a b^{8} f^{18} - b^{9} f^{18}\right )}} + \frac {5 \, {\left (3 \, a b + 4 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right )}{\sqrt {a b - b^{2}}}\right )}{8 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b - b^{2}} f} - \frac {\frac {9 \, a^{2} b \cos \left (f x + e\right )^{3}}{f} - \frac {5 \, a b^{2} \cos \left (f x + e\right )^{3}}{f} - \frac {4 \, b^{3} \cos \left (f x + e\right )^{3}}{f} + \frac {7 \, a b^{2} \cos \left (f x + e\right )}{f} + \frac {4 \, b^{3} \cos \left (f x + e\right )}{f}}{8 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.45, size = 1154, normalized size = 6.41 \begin {gather*} -\frac {\frac {16\,a^3+83\,a^2\,b+6\,a\,b^2}{12\,\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 )}^2\,\left (-8\,a^3+299\,a\,b^2+24\,b^3\right )}{6\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {5\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (4\,b^2+3\,a\,b\right )}{4\,\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 )}^{10}\,\left (8\,a^4-32\,a^3\,b+93\,a^2\,b^2+28\,a\,b^3+8\,b^4\right )}{2\,a\,\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 )}^6\,\left (56\,a^4-144\,a^3\,b+31\,a^2\,b^2+546\,a\,b^3+36\,b^4\right )}{3\,a\,\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 (-96\,a^4+71\,a^3\,b+344\,a^2\,b^2+1208\,a\,b^3+48\,b^4\right )}{12\,a\,\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 (-176\,a^4+569\,a^3\,b-666\,a^2\,b^2+1704\,a\,b^3+144\,b^4\right )}{12\,a\,\left (a-b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (8\,a\,b-a^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (8\,a\,b-a^2\right )+a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-3\,a^2+8\,a\,b+16\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (-3\,a^2+8\,a\,b+16\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (3\,a^2-16\,a\,b+48\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (3\,a^2-16\,a\,b+48\,b^2\right )+a^2\right )}-\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {2\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {5\,\sqrt {b}\,\left (3\,a+4\,b\right )\,\left (240\,a^{11}\,b-1600\,a^{10}\,b^2+4160\,a^9\,b^3-4480\,a^8\,b^4-1120\,a^7\,b^5+8960\,a^6\,b^6-11200\,a^5\,b^7+7040\,a^4\,b^8-2320\,a^3\,b^9+320\,a^2\,b^{10}\right )}{16\,a\,{\left (a-b\right )}^{17/2}}-\frac {25\,\sqrt {b}\,\left (a-2\,b\right )\,{\left (3\,a+4\,b\right )}^2\,\left (-128\,a^{15}+1792\,a^{14}\,b-11520\,a^{13}\,b^2+45056\,a^{12}\,b^3-119680\,a^{11}\,b^4+228096\,a^{10}\,b^5-321024\,a^9\,b^6+337920\,a^8\,b^7-266112\,a^7\,b^8+154880\,a^6\,b^9-64768\,a^5\,b^{10}+18432\,a^4\,b^{11}-3200\,a^3\,b^{12}+256\,a^2\,b^{13}\right )}{512\,a\,{\left (a-b\right )}^{27/2}}\right )-\frac {25\,\sqrt {b}\,\left (a-2\,b\right )\,{\left (3\,a+4\,b\right )}^2\,\left (128\,a^{15}-1536\,a^{14}\,b+8448\,a^{13}\,b^2-28160\,a^{12}\,b^3+63360\,a^{11}\,b^4-101376\,a^{10}\,b^5+118272\,a^9\,b^6-101376\,a^8\,b^7+63360\,a^7\,b^8-28160\,a^6\,b^9+8448\,a^5\,b^{10}-1536\,a^4\,b^{11}+128\,a^3\,b^{12}\right )}{512\,a\,{\left (a-b\right )}^{27/2}}\right )\,{\left (a-b\right )}^9}{225\,a^{12}\,b-1200\,a^{11}\,b^2+1900\,a^{10}\,b^3+1000\,a^9\,b^4-6650\,a^8\,b^5+7000\,a^7\,b^6+700\,a^6\,b^7-7400\,a^5\,b^8+6625\,a^4\,b^9-2600\,a^3\,b^{10}+400\,a^2\,b^{11}}\right )\,\left (3\,a+4\,b\right )}{8\,f\,{\left (a-b\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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