Optimal. Leaf size=468 \[ -\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \]
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Rubi [A]
time = 0.20, antiderivative size = 468, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4025, 186, 65,
212, 44} \begin {gather*} -\frac {\left (c^2-d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 c^2 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\sqrt {2} c^2 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {3 (c-d)^2 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 186
Rule 212
Rule 4025
Rubi steps
\begin {align*} \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^2}{x \sqrt {a-a x} (a+a 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 {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \left (\frac {c^2}{a^3 x \sqrt {a-a x}}-\frac {(c-d)^2}{a^3 (1+x)^3 \sqrt {a-a x}}+\frac {-c^2+d^2}{a^3 (1+x)^2 \sqrt {a-a x}}-\frac {c^2}{a^3 (1+x) \sqrt {a-a x}}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\left (c^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (c^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((c-d)^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x)^3 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (\left (c^2-d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 c^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, 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 {\left (2 c^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, 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 {\left (3 (c-d)^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{8 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (\left (c^2-d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (3 (c-d)^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{32 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (\left (c^2-d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{2 a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (3 (c-d)^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{16 a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 28.03, size = 16249, normalized size = 34.72 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1132\) vs.
\(2(403)=806\).
time = 1.58, size = 1133, normalized size = 2.42
method | result | size |
default | \(\text {Expression too large to display}\) | \(1133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 31.18, size = 833, normalized size = 1.78 \begin {gather*} \left [\frac {\sqrt {2} {\left ({\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 43 \, c^{2} - 6 \, c d - 5 \, d^{2} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 64 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 4 \, {\left ({\left (15 \, c^{2} - 14 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}, \frac {\sqrt {2} {\left ({\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 43 \, c^{2} - 6 \, c d - 5 \, d^{2} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 64 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 2 \, {\left ({\left (15 \, c^{2} - 14 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{32 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} 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*} \int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{2}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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