Optimal. Leaf size=43 \[ -\frac {2}{b d \sqrt {d \tan (a+b x)}}+\frac {2 (d \tan (a+b x))^{3/2}}{3 b d^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 43, 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 (a+b x))^{3/2}}{3 b d^3}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \frac {\sec ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{(d x)^{3/2}} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(d x)^{3/2}}+\frac {\sqrt {d x}}{d^2}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac {2}{b d \sqrt {d \tan (a+b x)}}+\frac {2 (d \tan (a+b x))^{3/2}}{3 b d^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 32, normalized size = 0.74 \begin {gather*} \frac {2 \left (-3+\tan ^2(a+b x)\right )}{3 b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 50, normalized size = 1.16
method | result | size |
default | \(-\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-1\right ) \sin \left (b x +a \right )}{3 b \cos \left (b x +a \right )^{3} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (\frac {3}{\sqrt {d \tan \left (b x + a\right )}} - \frac {\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}{d^{2}}\right )}}{3 \, b d} \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 = 1.26 \begin {gather*} -\frac {2 \, {\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{3 \, b d^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right )} \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 \frac {\sec ^{4}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 44, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {d \tan \left (b x + a\right )} \tan \left (b x + a\right )}{b d} - \frac {3}{\sqrt {d \tan \left (b x + a\right )} b}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.97, size = 64, normalized size = 1.49 \begin {gather*} -\frac {4\,\left (\sin \left (2\,a+2\,b\,x\right )+\sin \left (4\,a+4\,b\,x\right )\right )\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}}{3\,b\,d^2\,{\sin \left (2\,a+2\,b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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