Optimal. Leaf size=34 \[ \frac {2 (b \tan (e+f x))^{7/2}}{7 b f (d \sec (e+f x))^{7/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2605} \[ \frac {2 (b \tan (e+f x))^{7/2}}{7 b f (d \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2605
Rubi steps
\begin {align*} \int \frac {(b \tan (e+f x))^{5/2}}{(d \sec (e+f x))^{7/2}} \, dx &=\frac {2 (b \tan (e+f x))^{7/2}}{7 b f (d \sec (e+f x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 45, normalized size = 1.32 \[ \frac {2 b^2 \sin ^3(e+f x) \sqrt {b \tan (e+f x)}}{7 d^3 f \sqrt {d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 68, normalized size = 2.00 \[ -\frac {2 \, {\left (b^{2} \cos \left (f x + e\right )^{3} - b^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{7 \, d^{4} f} \]
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 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 50, normalized size = 1.47 \[ \frac {2 \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )}{7 f \cos \left (f x +e \right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 72, normalized size = 2.12 \[ \frac {b^2\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}\,\left (2\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{28\,d^4\,f} \]
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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