Optimal. Leaf size=182 \[ \frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 7, 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^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \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(c+d x)\right )^{5/2} \, dx &=\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^{10}(c+d x) \, dx\\ &=\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^8(c+d x) \, dx\\ &=-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^6(c+d x) \, dx\\ &=\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^4(c+d x) \, dx\\ &=-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 86, normalized size = 0.47 \begin {gather*} \frac {\cot (c+d x) \left (35-45 \cot ^2(c+d x)+63 \cot ^4(c+d x)-105 \cot ^6(c+d x)+315 \cot ^8(c+d x)-315 \text {ArcTan}(\tan (c+d x)) \cot ^9(c+d x)\right ) \left (b \tan ^4(c+d x)\right )^{5/2}}{315 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 84, normalized size = 0.46
method | result | size |
derivativedivides | \(-\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\) | \(84\) |
default | \(-\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\) | \(84\) |
risch | \(\frac {b^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, x}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 i b^{2} \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, \left (1575 \,{\mathrm e}^{16 i \left (d x +c \right )}+6300 \,{\mathrm e}^{14 i \left (d x +c \right )}+21000 \,{\mathrm e}^{12 i \left (d x +c \right )}+31500 \,{\mathrm e}^{10 i \left (d x +c \right )}+39438 \,{\mathrm e}^{8 i \left (d x +c \right )}+26292 \,{\mathrm e}^{6 i \left (d x +c \right )}+13968 \,{\mathrm e}^{4 i \left (d x +c \right )}+3492 \,{\mathrm e}^{2 i \left (d x +c \right )}+563\right )}{315 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7} d}\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 79, normalized size = 0.43 \begin {gather*} \frac {35 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{9} - 45 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{7} + 63 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{5} - 105 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{3} - 315 \, {\left (d x + c\right )} b^{\frac {5}{2}} + 315 \, b^{\frac {5}{2}} \tan \left (d x + c\right )}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 96, normalized size = 0.53 \begin {gather*} \frac {{\left (35 \, b^{2} \tan \left (d x + c\right )^{9} - 45 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, b^{2} \tan \left (d x + c\right )^{5} - 105 \, b^{2} \tan \left (d x + c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{4}}}{315 \, d \tan \left (d x + c\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 (c + d x \right )}\right )^{\frac {5}{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. 960 vs.
\(2 (162) = 324\).
time = 5.60, size = 960, normalized size = 5.27 \begin {gather*} -\frac {{\left (315 \, b^{2} d x \tan \left (d x\right )^{9} \tan \left (c\right )^{9} - 2835 \, b^{2} d x \tan \left (d x\right )^{8} \tan \left (c\right )^{8} + 315 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{8} + 315 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{9} + 11340 \, b^{2} d x \tan \left (d x\right )^{7} \tan \left (c\right )^{7} - 105 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{6} - 2835 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{7} - 2835 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{8} - 105 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{9} - 26460 \, b^{2} d x \tan \left (d x\right )^{6} \tan \left (c\right )^{6} + 63 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{4} + 945 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{5} + 11340 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{6} + 11340 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{7} + 945 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{8} + 63 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{9} + 39690 \, b^{2} d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 45 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{2} - 567 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{3} - 3780 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{4} - 26460 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{5} - 26460 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{6} - 3780 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{7} - 567 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{8} - 45 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{9} - 39690 \, b^{2} d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 35 \, b^{2} \tan \left (d x\right )^{9} + 405 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right ) + 2268 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{2} + 8820 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{3} + 39690 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{4} + 39690 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{5} + 8820 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{6} + 2268 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{7} + 405 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{8} + 35 \, b^{2} \tan \left (c\right )^{9} + 26460 \, b^{2} d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 45 \, b^{2} \tan \left (d x\right )^{7} - 567 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right ) - 3780 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{2} - 26460 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 26460 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{4} - 3780 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{5} - 567 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{6} - 45 \, b^{2} \tan \left (c\right )^{7} - 11340 \, b^{2} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 63 \, b^{2} \tan \left (d x\right )^{5} + 945 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right ) + 11340 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 11340 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 945 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{4} + 63 \, b^{2} \tan \left (c\right )^{5} + 2835 \, b^{2} d x \tan \left (d x\right ) \tan \left (c\right ) - 105 \, b^{2} \tan \left (d x\right )^{3} - 2835 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) - 2835 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} - 105 \, b^{2} \tan \left (c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x\right ) + 315 \, b^{2} \tan \left (c\right )\right )} \sqrt {b}}{315 \, {\left (d \tan \left (d x\right )^{9} \tan \left (c\right )^{9} - 9 \, d \tan \left (d x\right )^{8} \tan \left (c\right )^{8} + 36 \, d \tan \left (d x\right )^{7} \tan \left (c\right )^{7} - 84 \, d \tan \left (d x\right )^{6} \tan \left (c\right )^{6} + 126 \, d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 126 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 84 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 36 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 9 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \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 (c+d\,x\right )}^4\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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