Optimal. Leaf size=214 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} c^2 f}-\frac {9 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{8 \sqrt {2} a^{3/2} c^2 f}+\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f} \]
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Rubi [A]
time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3989, 3972,
483, 597, 536, 209} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{3/2} c^2 f}-\frac {9 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{8 \sqrt {2} a^{3/2} c^2 f}+\frac {\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f}+\frac {7 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{8 a^2 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 483
Rule 536
Rule 597
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2} \, dx &=\frac {\int \cot ^4(e+f x) \sqrt {a+a \sec (e+f x)} \, dx}{a^2 c^2}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^3 c^2 f}\\ &=-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}-\frac {\text {Subst}\left (\int \frac {-a-5 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{2 a^4 c^2 f}\\ &=\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}+\frac {\text {Subst}\left (\int \frac {21 a^2-3 a^3 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{12 a^4 c^2 f}\\ &=\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}-\frac {\text {Subst}\left (\int \frac {69 a^3+21 a^4 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{24 a^4 c^2 f}\\ &=\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}+\frac {9 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{8 a c^2 f}-\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a c^2 f}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} c^2 f}-\frac {9 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{8 \sqrt {2} a^{3/2} c^2 f}+\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac {\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 24.32, size = 5612, normalized size = 26.22 \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. \(386\) vs.
\(2(184)=368\).
time = 0.26, size = 387, normalized size = 1.81
method | result | size |
default | \(-\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (48 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}+27 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-48 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-27 \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right )-62 \left (\cos ^{3}\left (f x +e \right )\right )+4 \left (\cos ^{2}\left (f x +e \right )\right )+42 \cos \left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )-1\right )}{48 c^{2} f \sin \left (f x +e \right )^{5} a^{2}}\) | \(387\) |
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 = 2.99, size = 608, normalized size = 2.84 \begin {gather*} \left [-\frac {27 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} - 1\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 ) \sin \left (f x + e\right ) + 48 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) - 4 \, {\left (31 \, \cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right )^{2} - 21 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{96 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )}, \frac {27 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} - 1\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 ) \sin \left (f x + e\right ) + 48 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a} \arctan \left (\frac {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 )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (31 \, \cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right )^{2} - 21 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{48 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\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 {1}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )} - a \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} - a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c^{2}} \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 {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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