Optimal. Leaf size=45 \[ \frac {2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {2 (d \tan (e+f x))^{7/2}}{7 d^3 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, 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 = {2687, 14}
\begin {gather*} \frac {2 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac {2 (d \tan (e+f x))^{3/2}}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \sec ^4(e+f x) \sqrt {d \tan (e+f x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {d x} \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\sqrt {d x}+\frac {(d x)^{5/2}}{d^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {2 (d \tan (e+f x))^{7/2}}{7 d^3 f}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 34, normalized size = 0.76 \begin {gather*} \frac {2 \left (4+3 \sec ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{21 d f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 50, normalized size = 1.11
method | result | size |
default | \(\frac {2 \left (4 \left (\cos ^{2}\left (f x +e \right )\right )+3\right ) \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )}{21 f \cos \left (f x +e \right )^{3}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (3 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} + 7 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{2}\right )}}{21 \, d^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 54, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{21 \, f \cos \left (f x + e\right )^{3}} \end {gather*}
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 \begin {gather*} \int \sqrt {d \tan {\left (e + f x \right )}} \sec ^{4}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 57, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right )^{3} + 7 \, \sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right )\right )}}{21 \, d^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.09, size = 218, normalized size = 4.84 \begin {gather*} -\frac {\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}}\,8{}\mathrm {i}}{21\,f}-\frac {\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}}\,8{}\mathrm {i}}{21\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {\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 {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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