Optimal. Leaf size=67 \[ \frac {2 (d \tan (a+b x))^{13/2}}{13 b d^5}+\frac {4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac {2 (d \tan (a+b x))^{5/2}}{5 b d} \]
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Rubi [A] time = 0.06, antiderivative size = 67, 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 = {2607, 270} \[ \frac {2 (d \tan (a+b x))^{13/2}}{13 b d^5}+\frac {4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac {2 (d \tan (a+b x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rubi steps
\begin {align*} \int \sec ^6(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (d x)^{3/2} \left (1+x^2\right )^2 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left ((d x)^{3/2}+\frac {2 (d x)^{7/2}}{d^2}+\frac {(d x)^{11/2}}{d^4}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {2 (d \tan (a+b x))^{5/2}}{5 b d}+\frac {4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac {2 (d \tan (a+b x))^{13/2}}{13 b d^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 52, normalized size = 0.78 \[ \frac {2 d \left (45 \sec ^6(a+b x)-5 \sec ^4(a+b x)-8 \sec ^2(a+b x)-32\right ) \sqrt {d \tan (a+b x)}}{585 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 68, normalized size = 1.01 \[ -\frac {2 \, {\left (32 \, d \cos \left (b x + a\right )^{6} + 8 \, d \cos \left (b x + a\right )^{4} + 5 \, d \cos \left (b x + a\right )^{2} - 45 \, d\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{585 \, b \cos \left (b x + a\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 78, normalized size = 1.16 \[ \frac {2 \, {\left (45 \, \sqrt {d \tan \left (b x + a\right )} d^{6} \tan \left (b x + a\right )^{6} + 130 \, \sqrt {d \tan \left (b x + a\right )} d^{6} \tan \left (b x + a\right )^{4} + 117 \, \sqrt {d \tan \left (b x + a\right )} d^{6} \tan \left (b x + a\right )^{2}\right )}}{585 \, b d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 60, normalized size = 0.90 \[ \frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )+40 \left (\cos ^{2}\left (b x +a \right )\right )+45\right ) \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{585 b \cos \left (b x +a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 51, normalized size = 0.76 \[ \frac {2 \, {\left (45 \, \left (d \tan \left (b x + a\right )\right )^{\frac {13}{2}} + 130 \, \left (d \tan \left (b x + a\right )\right )^{\frac {9}{2}} d^{2} + 117 \, \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} d^{4}\right )}}{585 \, b d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.19, size = 392, normalized size = 5.85 \[ -\frac {64\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{585\,b}-\frac {64\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{585\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}-\frac {32\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{195\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {1216\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{117\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {3488\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{117\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {384\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{13\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {128\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{13\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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