Optimal. Leaf size=213 \[ \frac {a^2 (3 c-2 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f (c-d)^{3/2} (c+d)^{7/2}}+\frac {a^2 (3 c-2 d) \tan (e+f x)}{2 f (c-d) (c+d)^3 (c+d \sec (e+f x))}+\frac {(3 c-2 d) \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{6 f (c-d) (c+d)^2 (c+d \sec (e+f x))^2}-\frac {d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f \left (c^2-d^2\right ) (c+d \sec (e+f x))^3} \]
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Rubi [A] time = 0.28, antiderivative size = 268, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3987, 96, 94, 93, 205} \[ -\frac {a^3 (3 c-2 d) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{f (c-d)^{3/2} (c+d)^{7/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {a^2 (3 c-2 d) \tan (e+f x)}{2 f (c-d) (c+d)^3 (c+d \sec (e+f x))}+\frac {(3 c-2 d) \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{6 f (c-d) (c+d)^2 (c+d \sec (e+f x))^2}-\frac {d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f \left (c^2-d^2\right ) (c+d \sec (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 205
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x} (c+d x)^4} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}-\frac {\left (a^2 (3 c-2 d) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {(3 c-2 d) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sec (e+f x))^2}-\frac {\left (a^3 (3 c-2 d) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 (c+d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {(3 c-2 d) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {a^2 (3 c-2 d) \tan (e+f x)}{2 (c-d) (c+d)^3 f (c+d \sec (e+f x))}-\frac {\left (a^4 (3 c-2 d) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 (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 {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {(3 c-2 d) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {a^2 (3 c-2 d) \tan (e+f x)}{2 (c-d) (c+d)^3 f (c+d \sec (e+f x))}-\frac {\left (a^4 (3 c-2 d) \tan (e+f x)\right ) \operatorname {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 )}{(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 {a^3 (3 c-2 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)}{(c-d)^{3/2} (c+d)^{7/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {(3 c-2 d) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {a^2 (3 c-2 d) \tan (e+f x)}{2 (c-d) (c+d)^3 f (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 4.78, size = 211, normalized size = 0.99 \[ \frac {a^2 (c-d)^2 \left (24 (3 c-2 d) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left (\frac {(d-c) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )-2 \sqrt {c^2-d^2} \sin (e+f x) \left (12 c^3-5 c^2 d+6 \left (c^3+6 c^2 d-7 c d^2-2 d^3\right ) \cos (e+f x)+\left (12 c^3-7 c^2 d-6 c d^2-2 d^3\right ) \cos (2 (e+f x))+6 c d^2-22 d^3\right )\right )}{24 f (d-c)^3 (c+d)^3 \sqrt {c^2-d^2} (c \cos (e+f x)+d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1234, normalized size = 5.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.11, size = 420, normalized size = 1.97 \[ -\frac {\frac {3 \, {\left (3 \, a^{2} c - 2 \, a^{2} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{4} + 2 \, c^{3} d - 2 \, c d^{3} - d^{4}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {9 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 21 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 16 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 16 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 21 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{4} + 2 \, c^{3} d - 2 \, c d^{3} - d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.83, size = 228, normalized size = 1.07 \[ \frac {8 a^{2} \left (-\frac {\frac {\left (3 c -2 d \right ) \left (c -d \right ) \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{8 c^{3}+24 c^{2} d +24 c \,d^{2}+8 d^{3}}-\frac {\left (3 c -2 d \right ) \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 \left (c^{2}+2 c d +d^{2}\right )}+\frac {\left (5 c -6 d \right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{8 \left (c +d \right ) \left (c -d \right )}}{\left (\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c -\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d -c -d \right )^{3}}+\frac {\left (3 c -2 d \right ) \arctanh \left (\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c -d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{8 \left (c^{4}+2 c^{3} d -2 c \,d^{3}-d^{4}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{f} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.09, size = 286, normalized size = 1.34 \[ \frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (3\,a^2\,c^2-5\,a^2\,c\,d+2\,a^2\,d^2\right )}{{\left (c+d\right )}^3}-\frac {8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a^2\,c-2\,a^2\,d\right )}{3\,{\left (c+d\right )}^2}+\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,c-6\,d\right )}{\left (c+d\right )\,\left (c-d\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-3\,c^3-3\,c^2\,d+3\,c\,d^2+3\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-3\,c^3+3\,c^2\,d+3\,c\,d^2-3\,d^3\right )+3\,c\,d^2+3\,c^2\,d+c^3+d^3-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )\right )}+\frac {2\,a^2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {c-d}}{\sqrt {c+d}}\right )\,\left (\frac {3\,c}{2}-d\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}} \]
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 \[ a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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