Optimal. Leaf size=123 \[ \frac {a x}{c^2}+\frac {2 \left (b c^3-2 a c^2 d+a d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^2 (c-d)^{3/2} (c+d)^{3/2} f}-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))} \]
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Rubi [A]
time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4008, 4004,
3916, 2738, 214} \begin {gather*} -\frac {d (b c-a d) \tan (e+f x)}{c f \left (c^2-d^2\right ) (c+d \sec (e+f x))}+\frac {2 \left (-2 a c^2 d+a d^3+b c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}+\frac {a x}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4008
Rubi steps
\begin {align*} \int \frac {a+b \sec (e+f x)}{(c+d \sec (e+f x))^2} \, dx &=-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))}-\frac {\int \frac {-a \left (c^2-d^2\right )-c (b c-a d) \sec (e+f x)}{c+d \sec (e+f x)} \, dx}{c \left (c^2-d^2\right )}\\ &=\frac {a x}{c^2}-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {\left (c^2 (b c-a d)-a d \left (c^2-d^2\right )\right ) \int \frac {\sec (e+f x)}{c+d \sec (e+f x)} \, dx}{c^2 \left (c^2-d^2\right )}\\ &=\frac {a x}{c^2}-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {\left (c^2 (b c-a d)-a d \left (c^2-d^2\right )\right ) \int \frac {1}{1+\frac {c \cos (e+f x)}{d}} \, dx}{c^2 d \left (c^2-d^2\right )}\\ &=\frac {a x}{c^2}-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {\left (2 \left (c^2 (b c-a d)-a d \left (c^2-d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c}{d}+\left (1-\frac {c}{d}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{c^2 d \left (c^2-d^2\right ) f}\\ &=\frac {a x}{c^2}+\frac {2 \left (b c^3-2 a c^2 d+a d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^2 (c-d)^{3/2} (c+d)^{3/2} f}-\frac {d (b c-a d) \tan (e+f x)}{c \left (c^2-d^2\right ) f (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 155, normalized size = 1.26 \begin {gather*} \frac {-\frac {2 \left (b c^3+a d \left (-2 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+\frac {a d \left (c^2-d^2\right ) (e+f x)+a c \left (c^2-d^2\right ) (e+f x) \cos (e+f x)-c d (b c-a d) \sin (e+f x)}{d+c \cos (e+f x)}}{c^2 (c-d) (c+d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 168, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {\frac {2 a \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2}}+\frac {-\frac {2 d \left (a d -b c \right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {2 \left (2 a \,c^{2} d -a \,d^{3}-b \,c^{3}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{2}}}{f}\) | \(168\) |
default | \(\frac {\frac {2 a \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2}}+\frac {-\frac {2 d \left (a d -b c \right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {2 \left (2 a \,c^{2} d -a \,d^{3}-b \,c^{3}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{2}}}{f}\) | \(168\) |
risch | \(\frac {a x}{c^{2}}-\frac {2 i d \left (-a d +b c \right ) \left (d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}{c^{2} \left (c^{2}-d^{2}\right ) f \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a \,d^{3}}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,c^{2}}-\frac {c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) b}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a \,d^{3}}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,c^{2}}+\frac {c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) b}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}\) | \(583\) |
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. 256 vs.
\(2 (117) = 234\).
time = 4.42, size = 577, normalized size = 4.69 \begin {gather*} \left [\frac {2 \, {\left (a c^{5} - 2 \, a c^{3} d^{2} + a c d^{4}\right )} f x \cos \left (f x + e\right ) + 2 \, {\left (a c^{4} d - 2 \, a c^{2} d^{3} + a d^{5}\right )} f x - {\left (b c^{3} d - 2 \, a c^{2} d^{2} + a d^{4} + {\left (b c^{4} - 2 \, a c^{3} d + a c d^{3}\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 (b c^{4} d - a c^{3} d^{2} - b c^{2} d^{3} + a c d^{4}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} d - 2 \, c^{4} d^{3} + c^{2} d^{5}\right )} f\right )}}, \frac {{\left (a c^{5} - 2 \, a c^{3} d^{2} + a c d^{4}\right )} f x \cos \left (f x + e\right ) + {\left (a c^{4} d - 2 \, a c^{2} d^{3} + a d^{5}\right )} f x + {\left (b c^{3} d - 2 \, a c^{2} d^{2} + a d^{4} + {\left (b c^{4} - 2 \, a c^{3} d + a c d^{3}\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 (b c^{4} d - a c^{3} d^{2} - b c^{2} d^{3} + a c d^{4}\right )} \sin \left (f x + e\right )}{{\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} d - 2 \, c^{4} d^{3} + c^{2} d^{5}\right )} f}\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*} \int \frac {a + b \sec {\left (e + f x \right )}}{\left (c + d \sec {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 201, normalized size = 1.63 \begin {gather*} \frac {\frac {2 \, {\left (b c^{3} - 2 \, a c^{2} d + a 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^{4} - c^{2} d^{2}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {{\left (f x + e\right )} a}{c^{2}} + \frac {2 \, {\left (b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c^{3} - c d^{2}\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 )}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.33, size = 2500, normalized size = 20.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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