Optimal. Leaf size=72 \[ \frac {2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}+\frac {8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2692, 2685}
\begin {gather*} \frac {8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}+\frac {2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2685
Rule 2692
Rubi steps
\begin {align*} \int \frac {\sqrt {b \tan (e+f x)}}{(d \sec (e+f x))^{7/2}} \, dx &=\frac {2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}+\frac {4 \int \frac {\sqrt {b \tan (e+f x)}}{(d \sec (e+f x))^{3/2}} \, dx}{7 d^2}\\ &=\frac {2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}+\frac {8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 53, normalized size = 0.74 \begin {gather*} \frac {(19 \sin (e+f x)+3 \sin (3 (e+f x))) \sqrt {b \tan (e+f x)}}{42 d^3 f \sqrt {d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 62, normalized size = 0.86
method | result | size |
default | \(\frac {2 \left (3 \left (\cos ^{2}\left (f x +e \right )\right )+4\right ) \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )}{21 f \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \cos \left (f x +e \right )^{3}}\) | \(62\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 69, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} + 4 \, \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 )}{21 \, d^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.35, size = 69, normalized size = 0.96 \begin {gather*} \frac {\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}\,\left (22\,\sin \left (2\,e+2\,f\,x\right )+3\,\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}}}{84\,d^4\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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