Optimal. Leaf size=258 \[ \frac {5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )+d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f} \]
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Rubi [A]
time = 0.29, antiderivative size = 315, normalized size of antiderivative = 1.22, number of steps
used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4072, 100, 155,
158, 152, 65, 223, 209} \begin {gather*} -\frac {d \left (c^2+10 c d-12 d^2\right ) \tan (e+f x) (c+d \sec (e+f x))^2}{3 a^2 f}-\frac {d \tan (e+f x) \left (d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)+4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )\right )}{6 a^2 f}+\frac {(c-d) (c+10 d) \tan (e+f x) (c+d \sec (e+f x))^3}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {5 d^2 (2 c-d) \left (2 c^2-3 c d+2 d^2\right ) \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 155
Rule 158
Rule 209
Rule 223
Rule 4072
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^5}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^3 \left (-a^2 \left (c^2+6 c d-4 d^2\right )+3 a^2 (c-2 d) d x\right )}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-3 a^4 (11 c-10 d) d^2+3 a^4 d \left (c^2+10 c d-12 d^2\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{3 a^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x) \left (3 a^6 d^2 \left (31 c^2-50 c d+24 d^2\right )-3 a^6 d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{9 a^6 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )+d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}-\frac {\left (5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )+d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}+\frac {\left (5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )+d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}+\frac {\left (5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d \left (c^2+10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a^2 f}+\frac {(c-d) (c+10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right )+d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}\\ \end {align*}
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Mathematica [A]
time = 4.18, size = 446, normalized size = 1.73 \begin {gather*} \frac {240 d^2 \left (-4 c^3+8 c^2 d-7 c d^2+2 d^3\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (6 c^5+15 c^4 d-120 c^3 d^2+420 c^2 d^3-300 c d^4+104 d^5+\left (6 c^5+60 c^4 d-300 c^3 d^2+840 c^2 d^3-585 c d^4+190 d^5\right ) \cos (e+f x)+4 \left (2 c^5+5 c^4 d-40 c^3 d^2+130 c^2 d^3-95 c d^4+30 d^5\right ) \cos (2 (e+f x))+2 c^5 \cos (3 (e+f x))+20 c^4 d \cos (3 (e+f x))-100 c^3 d^2 \cos (3 (e+f x))+280 c^2 d^3 \cos (3 (e+f x))-215 c d^4 \cos (3 (e+f x))+66 d^5 \cos (3 (e+f x))+2 c^5 \cos (4 (e+f x))+5 c^4 d \cos (4 (e+f x))-40 c^3 d^2 \cos (4 (e+f x))+100 c^2 d^3 \cos (4 (e+f x))-80 c d^4 \cos (4 (e+f x))+24 d^5 \cos (4 (e+f x))\right ) \sec ^3(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{24 a^2 f (1+\cos (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 436, normalized size = 1.69 Too large to display
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. 836 vs.
\(2 (258) = 516\).
time = 0.29, size = 836, normalized size = 3.24 \begin {gather*} \frac {d^{5} {\left (\frac {4 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{2} - \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {a^{2} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \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 {30 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {30 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - 5 \, c d^{4} {\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 )} + 10 \, c^{2} d^{3} {\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^{3} d^{2} {\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 )} + \frac {5 \, c^{4} d {\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 {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 = 1.62, size = 472, normalized size = 1.83 \begin {gather*} \frac {15 \, {\left ({\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left ({\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, d^{5} + 2 \, {\left (2 \, c^{5} + 5 \, c^{4} d - 40 \, c^{3} d^{2} + 100 \, c^{2} d^{3} - 80 \, c d^{4} + 24 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + {\left (2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 280 \, c^{2} d^{3} - 215 \, c d^{4} + 66 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 6 \, {\left (10 \, c^{2} d^{3} - 5 \, c d^{4} + 2 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (15 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{5} + 2 \, a^{2} f \cos \left (f x + e\right )^{4} + a^{2} f \cos \left (f x + e\right )^{3}\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 {\int \frac {c^{5} \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{5} \sec ^{6}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{2} d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{3} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c^{4} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \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. 506 vs.
\(2 (247) = 494\).
time = 0.59, size = 506, normalized size = 1.96 \begin {gather*} \frac {\frac {15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (60 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 75 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 40 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, d^{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 )}^{3} a^{2}} - \frac {a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 10 \, a^{4} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, a^{4} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{4} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{4} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 90 \, a^{4} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 150 \, a^{4} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 105 \, a^{4} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, a^{4} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.97, size = 268, normalized size = 1.04 \begin {gather*} \frac {5\,d^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (2\,c-d\right )\,\left (2\,c^2-3\,c\,d+2\,d^2\right )}{a^2\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2\,{\left (c-d\right )}^5}{a^2}-\frac {5\,\left (c+d\right )\,{\left (c-d\right )}^4}{2\,a^2}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\left (c-d\right )}^5}{6\,a^2\,f}-\frac {\left (20\,c^2\,d^3-25\,c\,d^4+10\,d^5\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-40\,c^2\,d^3+40\,c\,d^4-\frac {40\,d^5}{3}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (20\,c^2\,d^3-15\,c\,d^4+6\,d^5\right )\,\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 )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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