Optimal. Leaf size=67 \[ \frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]
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Rubi [A] time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2607, 270} \[ \frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rubi steps
\begin {align*} \int \sec ^6(e+f x) (d \tan (e+f x))^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int (d x)^{5/2} \left (1+x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left ((d x)^{5/2}+\frac {2 (d x)^{9/2}}{d^2}+\frac {(d x)^{13/2}}{d^4}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 52, normalized size = 0.78 \[ \frac {2 (44 \cos (2 (e+f x))+4 \cos (4 (e+f x))+117) \sec ^4(e+f x) (d \tan (e+f x))^{7/2}}{1155 d f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 82, normalized size = 1.22 \[ -\frac {2 \, {\left (32 \, d^{2} \cos \left (f x + e\right )^{6} + 24 \, d^{2} \cos \left (f x + e\right )^{4} + 21 \, d^{2} \cos \left (f x + e\right )^{2} - 77 \, d^{2}\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{1155 \, f \cos \left (f x + e\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 84, normalized size = 1.25 \[ \frac {2 \, {\left (77 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{7} + 210 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{5} + 165 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{3}\right )}}{1155 \, d^{5} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 60, normalized size = 0.90 \[ \frac {2 \left (32 \left (\cos ^{4}\left (f x +e \right )\right )+56 \left (\cos ^{2}\left (f x +e \right )\right )+77\right ) \left (\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )}{1155 f \cos \left (f x +e \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 51, normalized size = 0.76 \[ \frac {2 \, {\left (77 \, \left (d \tan \left (f x + e\right )\right )^{\frac {15}{2}} + 210 \, \left (d \tan \left (f x + e\right )\right )^{\frac {11}{2}} d^{2} + 165 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} d^{4}\right )}}{1155 \, d^{5} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.29, size = 474, normalized size = 7.07 \[ \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}}\,64{}\mathrm {i}}{1155\,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}}\,64{}\mathrm {i}}{1155\,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}}\,32{}\mathrm {i}}{385\,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}}\,2432{}\mathrm {i}}{231\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}+\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}}\,1504{}\mathrm {i}}{33\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}-\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}}\,4288{}\mathrm {i}}{55\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\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}}\,896{}\mathrm {i}}{15\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^6}-\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}}\,256{}\mathrm {i}}{15\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^7} \]
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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