Optimal. Leaf size=22 \[ \frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2607, 32} \[ \frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int (d x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 (d \tan (e+f x))^{7/2}}{7 d f}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 22, normalized size = 1.00 \[ \frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 55, normalized size = 2.50 \[ -\frac {2 \, {\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{7 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 28, normalized size = 1.27 \[ \frac {2 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )^{3}}{7 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 19, normalized size = 0.86 \[ \frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7 d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 18, normalized size = 0.82 \[ \frac {2 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}}}{7 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.58, size = 230, normalized size = 10.45 \[ \frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,2{}\mathrm {i}}{7\,f}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,12{}\mathrm {i}}{7\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,24{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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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