Optimal. Leaf size=203 \[ -\frac {35 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{64 \sqrt {2} a^2 c^{5/2} f}-\frac {35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2 \sec (e+f x)+a^2\right ) (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3960, 3796, 3795, 203} \[ -\frac {35 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{64 \sqrt {2} a^2 c^{5/2} f}-\frac {35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2 \sec (e+f x)+a^2\right ) (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3795
Rule 3796
Rule 3960
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}} \, dx &=\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \int \frac {\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2}} \, dx}{6 a}\\ &=\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac {35 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{5/2}} \, dx}{12 a^2}\\ &=-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac {35 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}} \, dx}{32 a^2 c}\\ &=-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}+\frac {35 \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{128 a^2 c^2}\\ &=-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac {35 \operatorname {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{64 a^2 c^2 f}\\ &=-\frac {35 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{64 \sqrt {2} a^2 c^{5/2} f}-\frac {35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 2.33, size = 434, normalized size = 2.14 \[ \frac {\cot ^4(e+f x) \left (\frac {105 i \sqrt {2} \left (-1+e^{i (e+f x)}\right )^5 \left (1+e^{i (e+f x)}\right )^4 \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )}{\left (1+e^{2 i (e+f x)}\right )^{9/2}}-3648 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sin ^9(e+f x) \csc \left (\frac {1}{2} (e+f x)\right ) \csc ^5(2 (e+f x))-5504 \sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^9(e+f x) \csc ^5(2 (e+f x))-13312 \sin ^2\left (\frac {1}{2} (e+f x)\right ) \sin ^8(e+f x) \csc ^5(2 (e+f x))+2752 \cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right ) \sin ^5\left (\frac {1}{2} (e+f x)\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x)+256 \sin ^5\left (\frac {1}{2} (e+f x)\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x)+114 \cot \left (\frac {e}{2}\right ) \tan ^4(e+f x) \sec (e+f x)-192 \cot \left (\frac {e}{2}\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x)+192 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x)\right )}{384 a^2 c^2 f \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 487, normalized size = 2.40 \[ \left [-\frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \, {\left (43 \, \cos \left (f x + e\right )^{4} - 161 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )^{2} + 105 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{768 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )}, \frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (43 \, \cos \left (f x + e\right )^{4} - 161 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )^{2} + 105 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{384 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.03, size = 551, normalized size = 2.71 \[ \frac {\left (-1+\cos \left (f x +e \right )\right )^{3} \left (21 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}} \left (\cos ^{2}\left (f x +e \right )\right )+12 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}} \cos \left (f x +e \right )+15 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \left (\cos ^{2}\left (f x +e \right )\right )-9 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}}-30 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \cos \left (f x +e \right )-21 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right )+15 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}}+42 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \cos \left (f x +e \right )+35 \left (\cos ^{2}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}-21 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}}-70 \cos \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}-105 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-105 \left (\cos ^{2}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )+35 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+210 \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+210 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )-105 \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-105 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )\right )}{48 a^{2} f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{5} \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sec \left (f x + e\right )}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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