Optimal. Leaf size=123 \[ \frac {10 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3486, 3769, 3771, 2641} \[ \frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {10 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3486
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx &=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+a \int \frac {1}{(d \sec (e+f x))^{7/2}} \, dx\\ &=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {(5 a) \int \frac {1}{(d \sec (e+f x))^{3/2}} \, dx}{7 d^2}\\ &=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {(5 a) \int \sqrt {d \sec (e+f x)} \, dx}{21 d^4}\\ &=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {\left (5 a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{21 d^4}\\ &=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {10 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 94, normalized size = 0.76 \[ \frac {\sqrt {d \sec (e+f x)} \left (26 a \sin (2 (e+f x))+3 a \sin (4 (e+f x))+40 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )-12 b \cos (2 (e+f x))-3 b \cos (4 (e+f x))-9 b\right )}{84 d^4 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sec \left (f x + e\right )} {\left (b \tan \left (f x + e\right ) + a\right )}}{d^{4} \sec \left (f x + e\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.82, size = 190, normalized size = 1.54 \[ \frac {\frac {10 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) a}{21}+\frac {10 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, a}{21}-\frac {2 b \left (\cos ^{4}\left (f x +e \right )\right )}{7}+\frac {2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a}{7}+\frac {10 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a}{21}}{f \cos \left (f x +e \right )^{4} \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]
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 \[ \int \frac {a + b \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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