Optimal. Leaf size=287 \[ \frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) \tan (e+f x) (c+d \sec (e+f x))^2}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac {d^2 \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \tan (e+f x) \sec (e+f x)}{30 a^3 f}-\frac {2 d \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right ) \tan (e+f x)}{15 a^3 f}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}+\frac {(c-d) (2 c+11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{15 a f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.42, antiderivative size = 329, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3987, 98, 150, 147, 63, 217, 203} \[ \frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) \tan (e+f x) (c+d \sec (e+f x))^2}{15 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {d \tan (e+f x) \left (d \left (30 c^2 d+4 c^3+146 c d^2-195 d^3\right ) \sec (e+f x)+4 \left (72 c^2 d^2+15 c^3 d+2 c^4-180 c d^3+76 d^4\right )\right )}{30 a^3 f}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}+\frac {(c-d) (2 c+11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{15 a f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 150
Rule 203
Rule 217
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^5}{\sqrt {a-a x} (a+a x)^{7/2}} \, 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)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(c+d x)^3 \left (-a^2 (2 c-d) (c+4 d)+a^2 (2 c-7 d) d x\right )}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(c+d x)^2 \left (-a^4 \left (2 c^3+9 c^2 d+37 c d^2-33 d^3\right )+a^4 d \left (4 c^2+24 c d-43 d^2\right ) x\right )}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 a^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(c+d x) \left (-a^6 d^2 \left (2 c^2+165 c d-152 d^2\right )+a^6 d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{15 a^7 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}-\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{2 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}+\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}+\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname {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^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d^3 \left (20 c^2-30 c d+13 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^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}\\ \end {align*}
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Mathematica [A] time = 2.23, size = 439, normalized size = 1.53 \[ \frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \left (12 c^5 \cos (3 (e+f x))+7 c^5 \cos (4 (e+f x))+29 c^5+90 c^4 d \cos (3 (e+f x))+15 c^4 d \cos (4 (e+f x))+105 c^4 d+120 c^3 d^2 \cos (3 (e+f x))+20 c^3 d^2 \cos (4 (e+f x))+340 c^3 d^2-1020 c^2 d^3 \cos (3 (e+f x))-220 c^2 d^3 \cos (4 (e+f x))-1940 c^2 d^3+3 \left (12 c^5+90 c^4 d+120 c^3 d^2-1020 c^2 d^3+1910 c d^4-777 d^5\right ) \cos (e+f x)+6 \left (6 c^5+20 c^4 d+60 c^3 d^2-360 c^2 d^3+630 c d^4-261 d^5\right ) \cos (2 (e+f x))+1710 c d^4 \cos (3 (e+f x))+360 c d^4 \cos (4 (e+f x))+3420 c d^4-717 d^5 \cos (3 (e+f x))-152 d^5 \cos (4 (e+f x))-1354 d^5\right )-480 d^3 \left (20 c^2-30 c d+13 d^2\right ) \cos ^6\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{120 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 502, normalized size = 1.75 \[ \frac {15 \, {\left ({\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left ({\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (15 \, d^{5} + 2 \, {\left (7 \, c^{5} + 15 \, c^{4} d + 20 \, c^{3} d^{2} - 220 \, c^{2} d^{3} + 360 \, c d^{4} - 152 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (4 \, c^{5} + 30 \, c^{4} d + 40 \, c^{3} d^{2} - 340 \, c^{2} d^{3} + 570 \, c d^{4} - 239 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (4 \, c^{5} + 30 \, c^{4} d + 140 \, c^{3} d^{2} - 640 \, c^{2} d^{3} + 1170 \, c d^{4} - 479 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + 15 \, {\left (10 \, c d^{4} - 3 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{60 \, {\left (a^{3} f \cos \left (f x + e\right )^{5} + 3 \, a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + a^{3} f \cos \left (f x + e\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.79, size = 679, normalized size = 2.37 \[ -\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{4} d}{4 f \,a^{3}}+\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{3} d^{2}}{2 f \,a^{3}}-\frac {c^{5} \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 f \,a^{3}}+\frac {5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{4} d}{4 f \,a^{3}}-\frac {10 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right ) c^{2} d^{3}}{f \,a^{3}}+\frac {15 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right ) c \,d^{4}}{f \,a^{3}}-\frac {5 d^{4} c}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}-\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{2} d^{3}}{2 f \,a^{3}}+\frac {7 d^{5}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{5}}{4 f \,a^{3}}-\frac {31 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d^{5}}{4 f \,a^{3}}-\frac {35 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{2} d^{3}}{2 f \,a^{3}}+\frac {5 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c \,d^{4}}{2 f \,a^{3}}+\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c \,d^{4}}{4 f \,a^{3}}+\frac {10 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right ) c^{2} d^{3}}{f \,a^{3}}-\frac {15 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right ) c \,d^{4}}{f \,a^{3}}-\frac {d^{5}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}+\frac {d^{5}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}-\frac {13 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right ) d^{5}}{2 f \,a^{3}}+\frac {13 \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right ) d^{5}}{2 f \,a^{3}}+\frac {7 d^{5}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{5}}{20 f \,a^{3}}-\frac {2 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{5}}{3 f \,a^{3}}+\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{5}}{20 f \,a^{3}}+\frac {85 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c \,d^{4}}{4 f \,a^{3}}-\frac {5 d^{4} c}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}+\frac {5 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{3} d^{2}}{3 f \,a^{3}}-\frac {10 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{2} d^{3}}{3 f \,a^{3}}+\frac {5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{3} d^{2}}{2 f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 689, normalized size = 2.40 \[ -\frac {d^{5} {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} - 15 \, c d^{4} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \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 )}} + \frac {\frac {85 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 10 \, c^{2} d^{3} {\left (\frac {\frac {105 \, \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 {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} - \frac {10 \, c^{3} d^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {c^{5} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {15 \, c^{4} d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.87, size = 252, normalized size = 0.88 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,{\left (c-d\right )}^5}{2\,a^3}-\frac {15\,\left (c+d\right )\,{\left (c-d\right )}^4}{4\,a^3}+\frac {5\,{\left (c+d\right )}^2\,{\left (c-d\right )}^3}{2\,a^3}\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (10\,c\,d^4-5\,d^5\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (10\,c\,d^4-7\,d^5\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {{\left (c-d\right )}^5}{4\,a^3}-\frac {5\,\left (c+d\right )\,{\left (c-d\right )}^4}{12\,a^3}\right )}{f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\left (c-d\right )}^5}{20\,a^3\,f}+\frac {d^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (20\,c^2-30\,c\,d+13\,d^2\right )}{a^3\,f} \]
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 \[ \frac {\int \frac {c^{5} \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{5} \sec ^{6}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{2} d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{3} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c^{4} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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