Optimal. Leaf size=194 \[ \frac {3 a^2 b \sec ^8(c+d x)}{8 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {b^3 \tan ^4(c+d x)}{4 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{2 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 b^3 \tan ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3587, 710,
1824} \begin {gather*} \frac {a^3 \tan (c+d x)}{d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {3 a^2 b \sec ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d}+\frac {3 b^3 \tan ^8(c+d x)}{8 d}+\frac {b^3 \tan ^6(c+d x)}{2 d}+\frac {b^3 \tan ^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 710
Rule 1824
Rule 3587
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int (a+x)^3 \left (1+\frac {x^2}{b^2}\right )^3 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {3 a^2 b \sec ^8(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \left (1+\frac {x^2}{b^2}\right )^3 \left (-3 a^2 x+(a+x)^3\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {3 a^2 b \sec ^8(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \left (a^3+\frac {3 a \left (a^2+b^2\right ) x^2}{b^2}+x^3+\frac {3 a \left (a^2+3 b^2\right ) x^4}{b^4}+\frac {3 x^5}{b^2}+\frac {a \left (a^2+9 b^2\right ) x^6}{b^6}+\frac {3 x^7}{b^4}+\frac {3 a x^8}{b^6}+\frac {x^9}{b^6}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {3 a^2 b \sec ^8(c+d x)}{8 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {b^3 \tan ^4(c+d x)}{4 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{2 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 b^3 \tan ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A]
time = 2.14, size = 177, normalized size = 0.91 \begin {gather*} \frac {\frac {1}{4} \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^4-\frac {6}{5} a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^5+\frac {1}{2} \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{7} a \left (5 a^2+3 b^2\right ) (a+b \tan (c+d x))^7+\frac {3}{8} \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {2}{3} a (a+b \tan (c+d x))^9+\frac {1}{10} (a+b \tan (c+d x))^{10}}{b^7 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 219, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{4}}\right )+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(219\) |
default | \(\frac {b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{4}}\right )+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(219\) |
risch | \(-\frac {32 \left (10 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{3}-315 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-525 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-630 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}-126 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-105 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-378 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}-315 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+120 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+i a \,b^{2}+315 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+126 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-105 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-135 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-360 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(306\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 176, normalized size = 0.91 \begin {gather*} \frac {84 \, b^{3} \tan \left (d x + c\right )^{10} + 280 \, a b^{2} \tan \left (d x + c\right )^{9} + 315 \, {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{8} + 120 \, {\left (a^{3} + 9 \, a b^{2}\right )} \tan \left (d x + c\right )^{7} + 420 \, {\left (3 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{6} + 504 \, {\left (a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{5} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 210 \, {\left (9 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right ) + 840 \, {\left (a^{3} + a b^{2}\right )} \tan \left (d x + c\right )^{3}}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 150, normalized size = 0.77 \begin {gather*} \frac {84 \, b^{3} + 105 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{9} + 8 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} + 6 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, a b^{2} \cos \left (d x + c\right ) + 5 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{10}} \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 (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{8}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 220, normalized size = 1.13 \begin {gather*} \frac {84 \, b^{3} \tan \left (d x + c\right )^{10} + 280 \, a b^{2} \tan \left (d x + c\right )^{9} + 315 \, a^{2} b \tan \left (d x + c\right )^{8} + 315 \, b^{3} \tan \left (d x + c\right )^{8} + 120 \, a^{3} \tan \left (d x + c\right )^{7} + 1080 \, a b^{2} \tan \left (d x + c\right )^{7} + 1260 \, a^{2} b \tan \left (d x + c\right )^{6} + 420 \, b^{3} \tan \left (d x + c\right )^{6} + 504 \, a^{3} \tan \left (d x + c\right )^{5} + 1512 \, a b^{2} \tan \left (d x + c\right )^{5} + 1890 \, a^{2} b \tan \left (d x + c\right )^{4} + 210 \, b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right )^{3} + 840 \, a b^{2} \tan \left (d x + c\right )^{3} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 840 \, a^{3} \tan \left (d x + c\right )}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 175, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {3\,a^3}{5}+\frac {9\,a\,b^2}{5}\right )+{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (\frac {a^3}{7}+\frac {9\,a\,b^2}{7}\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (\frac {3\,a^2\,b}{2}+\frac {b^3}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{4}\right )+a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^{10}}{10}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^9}{3}+a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^2+b^2\right )+\frac {3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^8\,\left (a^2+b^2\right )}{8}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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