Optimal. Leaf size=188 \[ \frac {a^3 \tanh ^{-1}(\sin (e+f x))}{d^3 f}-\frac {a^3 \sqrt {c-d} \left (2 c^2+6 c d+7 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{d^3 (c+d)^{5/2} f}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))} \]
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Rubi [A]
time = 0.27, antiderivative size = 301, normalized size of antiderivative = 1.60, number of steps
used = 9, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {4072, 100, 154,
163, 65, 223, 209, 95, 211} \begin {gather*} \frac {a^4 \sqrt {c-d} \left (2 c^2+6 c d+7 d^2\right ) \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 )}{d^3 f (c+d)^{5/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a^4 \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 f (c+d)^2 (c+d \sec (e+f x))}-\frac {(c-d) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{2 d f (c+d) (c+d \sec (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 100
Rule 154
Rule 163
Rule 209
Rule 211
Rule 223
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))^3} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{5/2}}{\sqrt {a-a x} (c+d x)^3} \, 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) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}+\frac {(a \tan (e+f x)) \text {Subst}\left (\int \frac {\sqrt {a+a x} \left (a^3 (c-5 d)-2 a^3 (c+d) x\right )}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 d (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^5 d (c+7 d)-2 a^5 (c+d)^2 x}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 d^2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))}-\frac {\left (a^5 \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 )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (a^5 \left (2 c (c+d)^2-d^2 (c+7 d)\right ) \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 )}{2 d^3 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))}+\frac {\left (2 a^4 \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 )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (a^5 \left (2 c (c+d)^2-d^2 (c+7 d)\right ) \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 )}{d^3 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^4 \sqrt {c-d} \left (2 c^2+6 c d+7 d^2\right ) \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)}{d^3 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))}+\frac {\left (2 a^4 \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 )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^4 \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {a^4 \sqrt {c-d} \left (2 c^2+6 c d+7 d^2\right ) \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)}{d^3 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{2 d (c+d) f (c+d \sec (e+f x))^2}-\frac {a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 (c+d)^2 f (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.48, size = 393, normalized size = 2.09 \begin {gather*} \frac {a^3 (d+c \cos (e+f x)) \sec ^6\left (\frac {1}{2} (e+f x)\right ) (1+\sec (e+f x))^3 \left (-4 (d+c \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (d+c \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {4 \left (2 c^3+4 c^2 d+c d^2-7 d^3\right ) \text {ArcTan}\left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (d+c \cos (e+f x))^2 (i \cos (e)+\sin (e))}{(c+d)^2 \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {(c-d) d \sec (e) \left (\left (2 c^4+6 c^3 d+5 c^2 d^2+12 c d^3+2 d^4\right ) \sin (e)-c \left (d \left (7 c^2+18 c d+2 d^2\right ) \sin (f x)-d \left (c^2+6 c d+2 d^2\right ) \sin (2 e+f x)+c \left (2 c^2+6 c d+d^2\right ) \sin (e+2 f x)\right )\right )}{c^2 (c+d)^2}\right )}{32 d^3 f (c+d \sec (e+f x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.61, size = 227, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {16 a^{3} \left (-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 d^{3}}+\frac {\left (c -d \right ) \left (\frac {\frac {d \left (2 c^{2}+3 c d -5 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2}+4 c d +2 d^{2}}-\frac {d \left (2 c +7 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (2 c^{2}+6 c d +7 d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{8 d^{3}}+\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 d^{3}}\right )}{f}\) | \(227\) |
default | \(\frac {16 a^{3} \left (-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 d^{3}}+\frac {\left (c -d \right ) \left (\frac {\frac {d \left (2 c^{2}+3 c d -5 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2}+4 c d +2 d^{2}}-\frac {d \left (2 c +7 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (2 c^{2}+6 c d +7 d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{8 d^{3}}+\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 d^{3}}\right )}{f}\) | \(227\) |
risch | \(\frac {i a^{3} \left (-c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}-5 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+4 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+2 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-2 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-4 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-7 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+10 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+2 d^{5} {\mathrm e}^{2 i \left (f x +e \right )}-7 c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}-11 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}+16 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}+2 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}-2 c^{5}-4 c^{4} d +5 c^{3} d^{2}+c^{2} d^{3}\right )}{c^{2} f \left (c +d \right )^{2} d^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )^{2}}+\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}+\frac {3 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}+\frac {7 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right )}{2 \left (c +d \right )^{3} f d}-\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}-\frac {3 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}-\frac {7 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right )}{2 \left (c +d \right )^{3} f d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{d^{3} f}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{d^{3} f}\) | \(706\) |
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. 566 vs.
\(2 (182) = 364\).
time = 4.27, size = 1208, normalized size = 6.43 \begin {gather*} \left [\frac {{\left (2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} + {\left (2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \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 \, {\left (c^{2} + c d + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \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 (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4} + {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4} + {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (3 \, a^{3} c^{2} d^{2} + 3 \, a^{3} c d^{3} - 6 \, a^{3} d^{4} + {\left (2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{4} d^{3} + 2 \, c^{3} d^{4} + c^{2} d^{5}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right )} f\right )}}, -\frac {{\left (2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} + {\left (2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (d \cos \left (f x + e\right ) + c\right )} \sqrt {-\frac {c - d}{c + d}}}{{\left (c - d\right )} \sin \left (f x + e\right )}\right ) - {\left (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4} + {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + {\left (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4} + {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + a^{3} c^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + {\left (3 \, a^{3} c^{2} d^{2} + 3 \, a^{3} c d^{3} - 6 \, a^{3} d^{4} + {\left (2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{3} + 2 \, c^{3} d^{4} + c^{2} d^{5}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d^{5} + 2 \, c d^{6} + d^{7}\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*} a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{3} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx\right ) \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. 376 vs.
\(2 (175) = 350\).
time = 0.64, size = 376, normalized size = 2.00 \begin {gather*} \frac {\frac {a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{d^{3}} - \frac {a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{d^{3}} + \frac {{\left (2 \, a^{3} c^{3} + 4 \, a^{3} c^{2} d + a^{3} c d^{2} - 7 \, a^{3} d^{3}\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 (c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {2 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 7 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\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 )}^{2}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.50, size = 2500, normalized size = 13.30 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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