Optimal. Leaf size=138 \[ \frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4}{4 b^5 d}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5}{5 b^5 d}+\frac {\left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6}{3 b^5 d}-\frac {4 a (a+b \tan (c+d x))^7}{7 b^5 d}+\frac {(a+b \tan (c+d x))^8}{8 b^5 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 138, 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 = {3587, 711}
\begin {gather*} \frac {\left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6}{3 b^5 d}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5}{5 b^5 d}+\frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4}{4 b^5 d}+\frac {(a+b \tan (c+d x))^8}{8 b^5 d}-\frac {4 a (a+b \tan (c+d x))^7}{7 b^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rule 3587
Rubi steps
\begin {align*} \int \sec ^6(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 )^2 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\left (a^2+b^2\right )^2 (a+x)^3}{b^4}-\frac {4 a \left (a^2+b^2\right ) (a+x)^4}{b^4}+\frac {2 \left (3 a^2+b^2\right ) (a+x)^5}{b^4}-\frac {4 a (a+x)^6}{b^4}+\frac {(a+x)^7}{b^4}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4}{4 b^5 d}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5}{5 b^5 d}+\frac {\left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6}{3 b^5 d}-\frac {4 a (a+b \tan (c+d x))^7}{7 b^5 d}+\frac {(a+b \tan (c+d x))^8}{8 b^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 115, normalized size = 0.83 \begin {gather*} \frac {\frac {1}{4} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4-\frac {4}{5} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5+\frac {1}{3} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{7} a (a+b \tan (c+d x))^7+\frac {1}{8} (a+b \tan (c+d x))^8}{b^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 173, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2 \cos \left (d x +c \right )^{6}}+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(173\) |
default | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2 \cos \left (d x +c \right )^{6}}+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(173\) |
risch | \(-\frac {16 \left (-245 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-210 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+70 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-70 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+210 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-420 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-70 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-196 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-7 i a^{3}-210 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+70 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-56 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+105 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+3 i a \,b^{2}-322 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+84 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-42 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 142, normalized size = 1.03 \begin {gather*} \frac {105 \, b^{3} \tan \left (d x + c\right )^{8} + 360 \, a b^{2} \tan \left (d x + c\right )^{7} + 140 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \tan \left (d x + c\right )^{6} + 168 \, {\left (a^{3} + 6 \, a b^{2}\right )} \tan \left (d x + c\right )^{5} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 210 \, {\left (6 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right ) + 280 \, {\left (2 \, a^{3} + 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 = 0.39, size = 128, normalized size = 0.93 \begin {gather*} \frac {105 \, b^{3} + 140 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 45 \, a b^{2} \cos \left (d x + c\right ) + 3 \, {\left (7 \, a^{3} - 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 )^{8}} \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 ^{6}{\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 = 0.84, size = 166, normalized size = 1.20 \begin {gather*} \frac {105 \, b^{3} \tan \left (d x + c\right )^{8} + 360 \, a b^{2} \tan \left (d x + c\right )^{7} + 420 \, a^{2} b \tan \left (d x + c\right )^{6} + 280 \, b^{3} \tan \left (d x + c\right )^{6} + 168 \, a^{3} \tan \left (d x + c\right )^{5} + 1008 \, a b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a^{2} b \tan \left (d x + c\right )^{4} + 210 \, b^{3} \tan \left (d x + c\right )^{4} + 560 \, 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 = 3.60, size = 139, normalized size = 1.01 \begin {gather*} \frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {2\,a^3}{3}+a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {a^3}{5}+\frac {6\,a\,b^2}{5}\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (\frac {a^2\,b}{2}+\frac {b^3}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {3\,a^2\,b}{2}+\frac {b^3}{4}\right )+a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^8}{8}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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