Optimal. Leaf size=142 \[ \frac {128 c^4 \tan (e+f x)}{5 a f \sqrt {c-c \sec (e+f x)}}+\frac {32 c^3 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{5 a f}+\frac {12 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)} \]
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Rubi [A] time = 0.23, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3954, 3793, 3792} \[ \frac {128 c^4 \tan (e+f x)}{5 a f \sqrt {c-c \sec (e+f x)}}+\frac {32 c^3 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{5 a f}+\frac {12 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 3954
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx &=\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(6 c) \int \sec (e+f x) (c-c \sec (e+f x))^{5/2} \, dx}{a}\\ &=\frac {12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left (48 c^2\right ) \int \sec (e+f x) (c-c \sec (e+f x))^{3/2} \, dx}{5 a}\\ &=\frac {32 c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{5 a f}+\frac {12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left (64 c^3\right ) \int \sec (e+f x) \sqrt {c-c \sec (e+f x)} \, dx}{5 a}\\ &=\frac {128 c^4 \tan (e+f x)}{5 a f \sqrt {c-c \sec (e+f x)}}+\frac {32 c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{5 a f}+\frac {12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 86, normalized size = 0.61 \[ -\frac {c^3 (245 \cos (e+f x)+86 \cos (2 (e+f x))+91 \cos (3 (e+f x))+90) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt {c-c \sec (e+f x)}}{10 a f (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 88, normalized size = 0.62 \[ -\frac {2 \, {\left (91 \, c^{3} \cos \left (f x + e\right )^{3} + 43 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{5 \, a f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.66, size = 112, normalized size = 0.79 \[ -\frac {8 \, \sqrt {2} c^{3} {\left (\frac {5 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{a} - \frac {15 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c + 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{2} + c^{3}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} a}\right )}}{5 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.76, size = 83, normalized size = 0.58 \[ -\frac {2 \left (91 \left (\cos ^{3}\left (f x +e \right )\right )+43 \left (\cos ^{2}\left (f x +e \right )\right )-7 \cos \left (f x +e \right )+1\right ) \cos \left (f x +e \right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}}{5 a f \sin \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 163, normalized size = 1.15 \[ \frac {8 \, {\left (16 \, \sqrt {2} c^{\frac {7}{2}} - \frac {56 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {35 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {5 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{5 \, a f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.33, size = 164, normalized size = 1.15 \[ -\frac {2\,c^3\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,86{}\mathrm {i}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,245{}\mathrm {i}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,180{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,245{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,86{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,91{}\mathrm {i}+91{}\mathrm {i}\right )}{5\,a\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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