Optimal. Leaf size=288 \[ -\frac {2 d^3 (4 c+3 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{9/2} (c+d)^{3/2} f}+\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {(2 c-9 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac {\left (2 c^2-12 c d+45 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))} \]
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Rubi [A]
time = 0.34, antiderivative size = 325, normalized size of antiderivative = 1.13, number of steps
used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 105, 157,
12, 95, 211} \begin {gather*} \frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 f (c-d)^4 (c+d) \left (a^3 \sec (e+f x)+a^3\right )}+\frac {2 d^3 (4 c+3 d) \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 f (c-d)^{9/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a f (c-d)^3 (c+d) (a \sec (e+f x)+a)^2}+\frac {(c+6 d) \tan (e+f x)}{5 f (c-d)^2 (c+d) (a \sec (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 211
Rule 4072
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^2 (c+3 d)-3 a^2 d x}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^4 \left (2 c^2-8 c d-15 d^2\right )-2 a^4 d (c+6 d) x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{5 a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^6 (c+d) \left (2 c^2-12 c d+45 d^2\right )+a^6 d \left (2 c^2-10 c d-27 d^2\right ) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^6 (c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {15 a^8 d^3 (4 c+3 d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^9 (c-d)^3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (d^3 (4 c+3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{a (c-d)^3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (2 d^3 (4 c+3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {a-a \sec (e+f x)}}\right )}{a (c-d)^3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 (c-d)^{9/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.10, size = 1772, normalized size = 6.15 \begin {gather*} \frac {(4 c+3 d) \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^2 \sec ^5(e+f x) \left (\frac {16 i d^3 \text {ArcTan}\left (\sec \left (\frac {f x}{2}\right ) \left (\frac {\cos (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}-\frac {i \sin (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}\right ) \left (-i d \sin \left (\frac {f x}{2}\right )+i c \sin \left (e+\frac {f x}{2}\right )\right )\right ) \cos (e)}{\sqrt {c^2-d^2} f \sqrt {\cos (2 e)-i \sin (2 e)}}+\frac {16 d^3 \text {ArcTan}\left (\sec \left (\frac {f x}{2}\right ) \left (\frac {\cos (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}-\frac {i \sin (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}\right ) \left (-i d \sin \left (\frac {f x}{2}\right )+i c \sin \left (e+\frac {f x}{2}\right )\right )\right ) \sin (e)}{\sqrt {c^2-d^2} f \sqrt {\cos (2 e)-i \sin (2 e)}}\right )}{(-c+d)^4 (c+d) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sec (e) \sec ^5(e+f x) \left (-55 c^5 \sin \left (\frac {f x}{2}\right )+135 c^4 d \sin \left (\frac {f x}{2}\right )-20 c^3 d^2 \sin \left (\frac {f x}{2}\right )-810 c^2 d^3 \sin \left (\frac {f x}{2}\right )-450 c d^4 \sin \left (\frac {f x}{2}\right )+150 d^5 \sin \left (\frac {f x}{2}\right )+47 c^5 \sin \left (\frac {3 f x}{2}\right )-137 c^4 d \sin \left (\frac {3 f x}{2}\right )+88 c^3 d^2 \sin \left (\frac {3 f x}{2}\right )+812 c^2 d^3 \sin \left (\frac {3 f x}{2}\right )+690 c d^4 \sin \left (\frac {3 f x}{2}\right )+75 d^5 \sin \left (\frac {3 f x}{2}\right )-50 c^5 \sin \left (e-\frac {f x}{2}\right )+130 c^4 d \sin \left (e-\frac {f x}{2}\right )-10 c^3 d^2 \sin \left (e-\frac {f x}{2}\right )-1030 c^2 d^3 \sin \left (e-\frac {f x}{2}\right )-990 c d^4 \sin \left (e-\frac {f x}{2}\right )-150 d^5 \sin \left (e-\frac {f x}{2}\right )+50 c^5 \sin \left (e+\frac {f x}{2}\right )-130 c^4 d \sin \left (e+\frac {f x}{2}\right )+10 c^3 d^2 \sin \left (e+\frac {f x}{2}\right )+1030 c^2 d^3 \sin \left (e+\frac {f x}{2}\right )+765 c d^4 \sin \left (e+\frac {f x}{2}\right )-150 d^5 \sin \left (e+\frac {f x}{2}\right )-55 c^5 \sin \left (2 e+\frac {f x}{2}\right )+135 c^4 d \sin \left (2 e+\frac {f x}{2}\right )-20 c^3 d^2 \sin \left (2 e+\frac {f x}{2}\right )-810 c^2 d^3 \sin \left (2 e+\frac {f x}{2}\right )-675 c d^4 \sin \left (2 e+\frac {f x}{2}\right )-150 d^5 \sin \left (2 e+\frac {f x}{2}\right )-30 c^5 \sin \left (e+\frac {3 f x}{2}\right )+90 c^4 d \sin \left (e+\frac {3 f x}{2}\right )-60 c^3 d^2 \sin \left (e+\frac {3 f x}{2}\right )-360 c^2 d^3 \sin \left (e+\frac {3 f x}{2}\right )-30 c d^4 \sin \left (e+\frac {3 f x}{2}\right )+75 d^5 \sin \left (e+\frac {3 f x}{2}\right )+47 c^5 \sin \left (2 e+\frac {3 f x}{2}\right )-137 c^4 d \sin \left (2 e+\frac {3 f x}{2}\right )+88 c^3 d^2 \sin \left (2 e+\frac {3 f x}{2}\right )+812 c^2 d^3 \sin \left (2 e+\frac {3 f x}{2}\right )+525 c d^4 \sin \left (2 e+\frac {3 f x}{2}\right )-75 d^5 \sin \left (2 e+\frac {3 f x}{2}\right )-30 c^5 \sin \left (3 e+\frac {3 f x}{2}\right )+90 c^4 d \sin \left (3 e+\frac {3 f x}{2}\right )-60 c^3 d^2 \sin \left (3 e+\frac {3 f x}{2}\right )-360 c^2 d^3 \sin \left (3 e+\frac {3 f x}{2}\right )-195 c d^4 \sin \left (3 e+\frac {3 f x}{2}\right )-75 d^5 \sin \left (3 e+\frac {3 f x}{2}\right )+20 c^5 \sin \left (e+\frac {5 f x}{2}\right )-76 c^4 d \sin \left (e+\frac {5 f x}{2}\right )+106 c^3 d^2 \sin \left (e+\frac {5 f x}{2}\right )+346 c^2 d^3 \sin \left (e+\frac {5 f x}{2}\right )+219 c d^4 \sin \left (e+\frac {5 f x}{2}\right )+15 d^5 \sin \left (e+\frac {5 f x}{2}\right )-15 c^5 \sin \left (2 e+\frac {5 f x}{2}\right )+45 c^4 d \sin \left (2 e+\frac {5 f x}{2}\right )-30 c^3 d^2 \sin \left (2 e+\frac {5 f x}{2}\right )-90 c^2 d^3 \sin \left (2 e+\frac {5 f x}{2}\right )+75 c d^4 \sin \left (2 e+\frac {5 f x}{2}\right )+15 d^5 \sin \left (2 e+\frac {5 f x}{2}\right )+20 c^5 \sin \left (3 e+\frac {5 f x}{2}\right )-76 c^4 d \sin \left (3 e+\frac {5 f x}{2}\right )+106 c^3 d^2 \sin \left (3 e+\frac {5 f x}{2}\right )+346 c^2 d^3 \sin \left (3 e+\frac {5 f x}{2}\right )+144 c d^4 \sin \left (3 e+\frac {5 f x}{2}\right )-15 d^5 \sin \left (3 e+\frac {5 f x}{2}\right )-15 c^5 \sin \left (4 e+\frac {5 f x}{2}\right )+45 c^4 d \sin \left (4 e+\frac {5 f x}{2}\right )-30 c^3 d^2 \sin \left (4 e+\frac {5 f x}{2}\right )-90 c^2 d^3 \sin \left (4 e+\frac {5 f x}{2}\right )-15 d^5 \sin \left (4 e+\frac {5 f x}{2}\right )+7 c^5 \sin \left (2 e+\frac {7 f x}{2}\right )-27 c^4 d \sin \left (2 e+\frac {7 f x}{2}\right )+38 c^3 d^2 \sin \left (2 e+\frac {7 f x}{2}\right )+72 c^2 d^3 \sin \left (2 e+\frac {7 f x}{2}\right )+15 c d^4 \sin \left (2 e+\frac {7 f x}{2}\right )+15 c d^4 \sin \left (3 e+\frac {7 f x}{2}\right )+7 c^5 \sin \left (4 e+\frac {7 f x}{2}\right )-27 c^4 d \sin \left (4 e+\frac {7 f x}{2}\right )+38 c^3 d^2 \sin \left (4 e+\frac {7 f x}{2}\right )+72 c^2 d^3 \sin \left (4 e+\frac {7 f x}{2}\right )\right )}{120 c (-c+d)^4 (c+d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.40, size = 284, normalized size = 0.99 Too large to display
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. 832 vs.
\(2 (283) = 566\).
time = 3.52, size = 1725, normalized size = 5.99 \begin {gather*} \left [\frac {15 \, {\left (4 \, c d^{4} + 3 \, d^{5} + {\left (4 \, c^{2} d^{3} + 3 \, c d^{4}\right )} \cos \left (f x + e\right )^{4} + {\left (12 \, c^{2} d^{3} + 13 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (4 \, c^{2} d^{3} + 15 \, c d^{4} + 9 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6} + {\left (7 \, c^{6} - 27 \, c^{5} d + 31 \, c^{4} d^{2} + 99 \, c^{3} d^{3} - 23 \, c^{2} d^{4} - 72 \, c d^{5} - 15 \, d^{6}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{6} - 29 \, c^{5} d + 51 \, c^{4} d^{2} + 193 \, c^{3} d^{3} + 60 \, c^{2} d^{4} - 164 \, c d^{5} - 117 \, d^{6}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c^{6} - 6 \, c^{5} d + 5 \, c^{4} d^{2} + 147 \, c^{3} d^{3} + 164 \, c^{2} d^{4} - 141 \, c d^{5} - 171 \, d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\left ({\left (a^{3} c^{8} - 3 \, a^{3} c^{7} d + a^{3} c^{6} d^{2} + 5 \, a^{3} c^{5} d^{3} - 5 \, a^{3} c^{4} d^{4} - a^{3} c^{3} d^{5} + 3 \, a^{3} c^{2} d^{6} - a^{3} c d^{7}\right )} f \cos \left (f x + e\right )^{4} + {\left (3 \, a^{3} c^{8} - 8 \, a^{3} c^{7} d + 16 \, a^{3} c^{5} d^{3} - 10 \, a^{3} c^{4} d^{4} - 8 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c^{2} d^{6} - a^{3} d^{8}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} c^{8} - 8 \, a^{3} c^{6} d^{2} + 8 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 16 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c d^{7} - 3 \, a^{3} d^{8}\right )} f \cos \left (f x + e\right ) + {\left (a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right )} f\right )}}, -\frac {15 \, {\left (4 \, c d^{4} + 3 \, d^{5} + {\left (4 \, c^{2} d^{3} + 3 \, c d^{4}\right )} \cos \left (f x + e\right )^{4} + {\left (12 \, c^{2} d^{3} + 13 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (4 \, c^{2} d^{3} + 15 \, c d^{4} + 9 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) - {\left (2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6} + {\left (7 \, c^{6} - 27 \, c^{5} d + 31 \, c^{4} d^{2} + 99 \, c^{3} d^{3} - 23 \, c^{2} d^{4} - 72 \, c d^{5} - 15 \, d^{6}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{6} - 29 \, c^{5} d + 51 \, c^{4} d^{2} + 193 \, c^{3} d^{3} + 60 \, c^{2} d^{4} - 164 \, c d^{5} - 117 \, d^{6}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c^{6} - 6 \, c^{5} d + 5 \, c^{4} d^{2} + 147 \, c^{3} d^{3} + 164 \, c^{2} d^{4} - 141 \, c d^{5} - 171 \, d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left ({\left (a^{3} c^{8} - 3 \, a^{3} c^{7} d + a^{3} c^{6} d^{2} + 5 \, a^{3} c^{5} d^{3} - 5 \, a^{3} c^{4} d^{4} - a^{3} c^{3} d^{5} + 3 \, a^{3} c^{2} d^{6} - a^{3} c d^{7}\right )} f \cos \left (f x + e\right )^{4} + {\left (3 \, a^{3} c^{8} - 8 \, a^{3} c^{7} d + 16 \, a^{3} c^{5} d^{3} - 10 \, a^{3} c^{4} d^{4} - 8 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c^{2} d^{6} - a^{3} d^{8}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} c^{8} - 8 \, a^{3} c^{6} d^{2} + 8 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 16 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c d^{7} - 3 \, a^{3} d^{8}\right )} f \cos \left (f x + e\right ) + {\left (a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right )} f\right )}}\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 {\sec {\left (e + f x \right )}}{c^{2} \sec ^{3}{\left (e + f x \right )} + 3 c^{2} \sec ^{2}{\left (e + f x \right )} + 3 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{4}{\left (e + f x \right )} + 6 c d \sec ^{3}{\left (e + f x \right )} + 6 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{5}{\left (e + f x \right )} + 3 d^{2} \sec ^{4}{\left (e + f x \right )} + 3 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \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. 918 vs.
\(2 (271) = 542\).
time = 0.58, size = 918, normalized size = 3.19 \begin {gather*} -\frac {\frac {120 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}} + \frac {120 \, {\left (4 \, c d^{3} + 3 \, d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {3 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a^{12} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 84 \, a^{12} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 168 \, a^{12} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 210 \, a^{12} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 168 \, a^{12} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 84 \, a^{12} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a^{12} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a^{12} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, a^{12} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 420 \, a^{12} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 980 \, a^{12} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1400 \, a^{12} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1260 \, a^{12} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 700 \, a^{12} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 220 \, a^{12} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 30 \, a^{12} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 180 \, a^{12} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1020 \, a^{12} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3180 \, a^{12} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5850 \, a^{12} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6540 \, a^{12} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4380 \, a^{12} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1620 \, a^{12} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 255 \, a^{12} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{10} - 10 \, a^{15} c^{9} d + 45 \, a^{15} c^{8} d^{2} - 120 \, a^{15} c^{7} d^{3} + 210 \, a^{15} c^{6} d^{4} - 252 \, a^{15} c^{5} d^{5} + 210 \, a^{15} c^{4} d^{6} - 120 \, a^{15} c^{3} d^{7} + 45 \, a^{15} c^{2} d^{8} - 10 \, a^{15} c d^{9} + a^{15} d^{10}}}{60 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.12, size = 464, normalized size = 1.61 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,a^3\,f\,{\left (c-d\right )}^2}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2\,\left (c^2-d^2\right )\,\left (\frac {1}{a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{2\,a^3\,{\left (c-d\right )}^4}\right )}{{\left (c-d\right )}^2}-\frac {3}{2\,a^3\,{\left (c-d\right )}^2}+\frac {{\left (c+d\right )}^2}{4\,a^3\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {1}{3\,a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{6\,a^3\,{\left (c-d\right )}^4}\right )}{f}+\frac {2\,d^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^3\,c^5-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^3\,c^5-5\,a^3\,c^4\,d+10\,a^3\,c^3\,d^2-10\,a^3\,c^2\,d^3+5\,a^3\,c\,d^4-a^3\,d^5\right )+a^3\,d^5-3\,a^3\,c\,d^4-3\,a^3\,c^4\,d+2\,a^3\,c^2\,d^3+2\,a^3\,c^3\,d^2\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^5-5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4\,d+10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d^2-10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^3+5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^5}{\sqrt {c+d}\,{\left (c-d\right )}^{9/2}}\right )\,\left (4\,c+3\,d\right )\,2{}\mathrm {i}}{a^3\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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