Optimal. Leaf size=136 \[ \frac {c^5 x}{a^2}-\frac {47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {13 c^5 \tan (e+f x)}{2 a^2 f}+\frac {112 c^5 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac {32 c^5 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (c^5-c^5 \sec (e+f x)\right ) \tan (e+f x)}{2 a^2 f} \]
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Rubi [A]
time = 0.28, antiderivative size = 153, normalized size of antiderivative = 1.12, number
of steps used = 26, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules
used = {3989, 3971, 3554, 8, 2686, 2687, 30, 3852, 2701, 308, 213, 2700, 276, 294}
\begin {gather*} \frac {7 c^5 \tan (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}-\frac {47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {c^5 x}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 213
Rule 276
Rule 294
Rule 308
Rule 2686
Rule 2687
Rule 2700
Rule 2701
Rule 3554
Rule 3852
Rule 3971
Rule 3989
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx &=\frac {\int \cot ^4(e+f x) (c-c \sec (e+f x))^7 \, dx}{a^2 c^2}\\ &=\frac {\int \left (c^7 \cot ^4(e+f x)-7 c^7 \cot ^3(e+f x) \csc (e+f x)+21 c^7 \cot ^2(e+f x) \csc ^2(e+f x)-35 c^7 \cot (e+f x) \csc ^3(e+f x)+35 c^7 \csc ^4(e+f x)-21 c^7 \csc ^4(e+f x) \sec (e+f x)+7 c^7 \csc ^4(e+f x) \sec ^2(e+f x)-c^7 \csc ^4(e+f x) \sec ^3(e+f x)\right ) \, dx}{a^2 c^2}\\ &=\frac {c^5 \int \cot ^4(e+f x) \, dx}{a^2}-\frac {c^5 \int \csc ^4(e+f x) \sec ^3(e+f x) \, dx}{a^2}-\frac {\left (7 c^5\right ) \int \cot ^3(e+f x) \csc (e+f x) \, dx}{a^2}+\frac {\left (7 c^5\right ) \int \csc ^4(e+f x) \sec ^2(e+f x) \, dx}{a^2}+\frac {\left (21 c^5\right ) \int \cot ^2(e+f x) \csc ^2(e+f x) \, dx}{a^2}-\frac {\left (21 c^5\right ) \int \csc ^4(e+f x) \sec (e+f x) \, dx}{a^2}-\frac {\left (35 c^5\right ) \int \cot (e+f x) \csc ^3(e+f x) \, dx}{a^2}+\frac {\left (35 c^5\right ) \int \csc ^4(e+f x) \, dx}{a^2}\\ &=-\frac {c^5 \cot ^3(e+f x)}{3 a^2 f}-\frac {c^5 \int \cot ^2(e+f x) \, dx}{a^2}+\frac {c^5 \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int x^2 \, dx,x,-\cot (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (35 c^5\right ) \text {Subst}\left (\int x^2 \, dx,x,\csc (e+f x)\right )}{a^2 f}-\frac {\left (35 c^5\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{a^2 f}\\ &=-\frac {34 c^5 \cot (e+f x)}{a^2 f}-\frac {19 c^5 \cot ^3(e+f x)}{a^2 f}-\frac {7 c^5 \csc (e+f x)}{a^2 f}+\frac {14 c^5 \csc ^3(e+f x)}{a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {c^5 \int 1 \, dx}{a^2}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{2 a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f}\\ &=\frac {c^5 x}{a^2}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {14 c^5 \csc (e+f x)}{a^2 f}+\frac {21 c^5 \csc ^3(e+f x)}{a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{2 a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}\\ &=\frac {c^5 x}{a^2}-\frac {21 c^5 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{2 a^2 f}\\ &=\frac {c^5 x}{a^2}-\frac {47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(136)=272\).
time = 3.44, size = 384, normalized size = 2.82 \begin {gather*} \frac {\cos ^3(e+f x) \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^6\left (\frac {1}{2} (e+f x)\right ) (c-c \sec (e+f x))^5 \left (-\frac {320 \cot ^2\left (\frac {1}{2} (e+f x)\right ) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )}{f}-\frac {64 \csc ^3\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )}{f}+3 \cot ^3\left (\frac {1}{2} (e+f x)\right ) \left (-4 x-\frac {94 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {94 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {1}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {1}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {28 \sin (f x)}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right )-\frac {64 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {e}{2}\right )}{f}\right )}{96 a^2 (1+\sec (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 137, normalized size = 1.01
method | result | size |
derivativedivides | \(\frac {16 c^{5} \left (\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{32}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{32}\right )}{f \,a^{2}}\) | \(137\) |
default | \(\frac {16 c^{5} \left (\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{32}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{32}\right )}{f \,a^{2}}\) | \(137\) |
risch | \(\frac {c^{5} x}{a^{2}}+\frac {i c^{5} \left (99 \,{\mathrm e}^{6 i \left (f x +e \right )}+435 \,{\mathrm e}^{5 i \left (f x +e \right )}+484 \,{\mathrm e}^{4 i \left (f x +e \right )}+930 \,{\mathrm e}^{3 i \left (f x +e \right )}+575 \,{\mathrm e}^{2 i \left (f x +e \right )}+507 \,{\mathrm e}^{i \left (f x +e \right )}+202\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}+\frac {47 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{2} f}-\frac {47 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{2} f}\) | \(164\) |
norman | \(\frac {\frac {c^{5} x}{a}+\frac {c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {6 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {45 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {491 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {641 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {111 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {32 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {16 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4} a}+\frac {47 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{2} f}-\frac {47 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{2} f}\) | \(285\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs.
\(2 (135) = 270\).
time = 0.49, size = 653, normalized size = 4.80 \begin {gather*} \frac {c^{5} {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} + 5 \, c^{5} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 10 \, c^{5} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - c^{5} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \frac {10 \, c^{5} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {5 \, c^{5} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.31, size = 260, normalized size = 1.91 \begin {gather*} \frac {12 \, c^{5} f x \cos \left (f x + e\right )^{4} + 24 \, c^{5} f x \cos \left (f x + e\right )^{3} + 12 \, c^{5} f x \cos \left (f x + e\right )^{2} - 141 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + 141 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (202 \, c^{5} \cos \left (f x + e\right )^{3} + 305 \, c^{5} \cos \left (f x + e\right )^{2} + 36 \, c^{5} \cos \left (f x + e\right ) - 3 \, c^{5}\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {c^{5} \left (\int \frac {5 \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {10 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {5 \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {1}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 153, normalized size = 1.12 \begin {gather*} \frac {\frac {6 \, {\left (f x + e\right )} c^{5}}{a^{2}} - \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 13 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} + \frac {32 \, {\left (a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.50, size = 145, normalized size = 1.07 \begin {gather*} \frac {c^5\,x}{a^2}-\frac {15\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-13\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^2\right )}+\frac {32\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}+\frac {16\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}-\frac {47\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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