Optimal. Leaf size=110 \[ \frac {b \cot (e+f x) \sqrt {b \tan ^4(e+f x)}}{f}-b x \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}-\frac {b \tan (e+f x) \sqrt {b \tan ^4(e+f x)}}{3 f}+\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 8}
\begin {gather*} -\frac {b \tan (e+f x) \sqrt {b \tan ^4(e+f x)}}{3 f}+\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f}-b x \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}+\frac {b \cot (e+f x) \sqrt {b \tan ^4(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
Rubi steps
\begin {align*} \int \left (b \tan ^4(e+f x)\right )^{3/2} \, dx &=\left (b \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}\right ) \int \tan ^6(e+f x) \, dx\\ &=\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f}-\left (b \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}\right ) \int \tan ^4(e+f x) \, dx\\ &=-\frac {b \tan (e+f x) \sqrt {b \tan ^4(e+f x)}}{3 f}+\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f}+\left (b \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}\right ) \int \tan ^2(e+f x) \, dx\\ &=\frac {b \cot (e+f x) \sqrt {b \tan ^4(e+f x)}}{f}-\frac {b \tan (e+f x) \sqrt {b \tan ^4(e+f x)}}{3 f}+\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f}-\left (b \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}\right ) \int 1 \, dx\\ &=\frac {b \cot (e+f x) \sqrt {b \tan ^4(e+f x)}}{f}-b x \cot ^2(e+f x) \sqrt {b \tan ^4(e+f x)}-\frac {b \tan (e+f x) \sqrt {b \tan ^4(e+f x)}}{3 f}+\frac {b \tan ^3(e+f x) \sqrt {b \tan ^4(e+f x)}}{5 f}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 66, normalized size = 0.60 \begin {gather*} \frac {\cot (e+f x) \left (3-5 \cot ^2(e+f x)+15 \cot ^4(e+f x)-15 \text {ArcTan}(\tan (e+f x)) \cot ^5(e+f x)\right ) \left (b \tan ^4(e+f x)\right )^{3/2}}{15 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 64, normalized size = 0.58
method | result | size |
derivativedivides | \(-\frac {\left (b \left (\tan ^{4}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (-3 \left (\tan ^{5}\left (f x +e \right )\right )+5 \left (\tan ^{3}\left (f x +e \right )\right )+15 \arctan \left (\tan \left (f x +e \right )\right )-15 \tan \left (f x +e \right )\right )}{15 f \tan \left (f x +e \right )^{6}}\) | \(64\) |
default | \(-\frac {\left (b \left (\tan ^{4}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (-3 \left (\tan ^{5}\left (f x +e \right )\right )+5 \left (\tan ^{3}\left (f x +e \right )\right )+15 \arctan \left (\tan \left (f x +e \right )\right )-15 \tan \left (f x +e \right )\right )}{15 f \tan \left (f x +e \right )^{6}}\) | \(64\) |
risch | \(\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}}\, x}{\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {2 i b \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}}\, \left (45 \,{\mathrm e}^{8 i \left (f x +e \right )}+90 \,{\mathrm e}^{6 i \left (f x +e \right )}+140 \,{\mathrm e}^{4 i \left (f x +e \right )}+70 \,{\mathrm e}^{2 i \left (f x +e \right )}+23\right )}{15 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3} f}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 57, normalized size = 0.52 \begin {gather*} \frac {3 \, b^{\frac {3}{2}} \tan \left (f x + e\right )^{5} - 5 \, b^{\frac {3}{2}} \tan \left (f x + e\right )^{3} - 15 \, {\left (f x + e\right )} b^{\frac {3}{2}} + 15 \, b^{\frac {3}{2}} \tan \left (f x + e\right )}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.14, size = 67, normalized size = 0.61 \begin {gather*} \frac {{\left (3 \, b \tan \left (f x + e\right )^{5} - 5 \, b \tan \left (f x + e\right )^{3} - 15 \, b f x + 15 \, b \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{4}}}{15 \, f \tan \left (f x + e\right )^{2}} \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 \left (b \tan ^{4}{\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1079 vs.
\(2 (106) = 212\).
time = 2.41, size = 1079, normalized size = 9.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^4\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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