Optimal. Leaf size=185 \[ \frac {3 \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^7 d}-\frac {a \left (10 a^2+9 b^2\right ) \tan (c+d x)}{b^6 d}+\frac {3 \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^5 d}-\frac {a \tan ^3(c+d x)}{b^4 d}+\frac {\tan ^4(c+d x)}{4 b^3 d}-\frac {\left (a^2+b^2\right )^3}{2 b^7 d (a+b \tan (c+d x))^2}+\frac {6 a \left (a^2+b^2\right )^2}{b^7 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.13, antiderivative size = 185, 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 {6 a \left (a^2+b^2\right )^2}{b^7 d (a+b \tan (c+d x))}-\frac {\left (a^2+b^2\right )^3}{2 b^7 d (a+b \tan (c+d x))^2}+\frac {3 \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^7 d}-\frac {a \left (10 a^2+9 b^2\right ) \tan (c+d x)}{b^6 d}+\frac {3 \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^5 d}-\frac {a \tan ^3(c+d x)}{b^4 d}+\frac {\tan ^4(c+d x)}{4 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rule 3587
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^3}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {-10 a^3-9 a b^2}{b^6}+\frac {3 \left (2 a^2+b^2\right ) x}{b^6}-\frac {3 a x^2}{b^6}+\frac {x^3}{b^6}+\frac {\left (a^2+b^2\right )^3}{b^6 (a+x)^3}-\frac {6 a \left (a^2+b^2\right )^2}{b^6 (a+x)^2}+\frac {3 \left (5 a^4+6 a^2 b^2+b^4\right )}{b^6 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {3 \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^7 d}-\frac {a \left (10 a^2+9 b^2\right ) \tan (c+d x)}{b^6 d}+\frac {3 \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^5 d}-\frac {a \tan ^3(c+d x)}{b^4 d}+\frac {\tan ^4(c+d x)}{4 b^3 d}-\frac {\left (a^2+b^2\right )^3}{2 b^7 d (a+b \tan (c+d x))^2}+\frac {6 a \left (a^2+b^2\right )^2}{b^7 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 5.38, size = 302, normalized size = 1.63 \begin {gather*} \frac {b^4 \sec ^4(c+d x) (a+b \tan (c+d x))^2-4 (a+b \tan (c+d x)) \left (3 a \left (5 a^4+6 a^2 b^2+b^4\right ) (\log (\cos (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+b \left (15 a^4+18 a^2 b^2+5 b^4+3 \left (5 a^4+6 a^2 b^2+b^4\right ) \log (\cos (c+d x))-3 \left (5 a^4+6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))\right ) \tan (c+d x)+2 a b^2 \left (5 a^2+4 b^2\right ) \tan ^2(c+d x)\right )+2 b^2 \sec ^2(c+d x) \left (5 a^4-b^4+2 a b \left (5 a^2+2 b^2\right ) \tan (c+d x)+2 b^2 \left (a^2+b^2\right ) \tan ^2(c+d x)-2 a b^3 \tan ^3(c+d x)\right )}{4 b^7 d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 195, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b^{3}}{4}+a \left (\tan ^{3}\left (d x +c \right )\right ) b^{2}-3 \left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b -\frac {3 b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+10 a^{3} \tan \left (d x +c \right )+9 b^{2} a \tan \left (d x +c \right )}{b^{6}}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{2 b^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {6 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (15 a^{4}+18 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(195\) |
default | \(\frac {-\frac {-\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b^{3}}{4}+a \left (\tan ^{3}\left (d x +c \right )\right ) b^{2}-3 \left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b -\frac {3 b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+10 a^{3} \tan \left (d x +c \right )+9 b^{2} a \tan \left (d x +c \right )}{b^{6}}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{2 b^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {6 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (15 a^{4}+18 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(195\) |
risch | \(\frac {30 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}+36 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-60 a^{4} b -52 a^{2} b^{3}-240 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-60 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}-188 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-32 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-172 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-210 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+30 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}+150 i a^{5} {\mathrm e}^{8 i \left (d x +c \right )}+300 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}+300 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+150 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+60 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}+72 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+6 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+12 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}+6 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}-4 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-4 i a^{3} b^{2}-26 i a \,b^{4}+30 i a^{5}+36 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+180 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+6 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+30 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+320 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+52 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+240 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-4 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-58 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} b^{6} d}+\frac {15 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{4}}{b^{7} d}+\frac {18 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{b^{5} d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{3} d}-\frac {15 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{4}}{b^{7} d}-\frac {18 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{5} d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\) | \(752\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 200, normalized size = 1.08 \begin {gather*} \frac {\frac {2 \, {\left (11 \, a^{6} + 21 \, a^{4} b^{2} + 9 \, a^{2} b^{4} - b^{6} + 12 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )}}{b^{9} \tan \left (d x + c\right )^{2} + 2 \, a b^{8} \tan \left (d x + c\right ) + a^{2} b^{7}} + \frac {b^{3} \tan \left (d x + c\right )^{4} - 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, {\left (2 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (10 \, a^{3} + 9 \, a b^{2}\right )} \tan \left (d x + c\right )}{b^{6}} + \frac {12 \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{7}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs.
\(2 (179) = 358\).
time = 0.47, size = 476, normalized size = 2.57 \begin {gather*} \frac {8 \, {\left (15 \, a^{4} b^{2} + 13 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} + b^{6} - 2 \, {\left (45 \, a^{4} b^{2} + 44 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left ({\left (5 \, a^{6} + a^{4} b^{2} - 5 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (5 \, a^{5} b + 6 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + {\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 6 \, {\left ({\left (5 \, a^{6} + a^{4} b^{2} - 5 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (5 \, a^{5} b + 6 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + {\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) - 2 \, {\left (a b^{5} \cos \left (d x + c\right ) + 2 \, {\left (15 \, a^{5} b - 2 \, a^{3} b^{3} - 13 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (2 \, a b^{8} d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + b^{9} d \cos \left (d x + c\right )^{4} + {\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{6}\right )}} \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 \frac {\sec ^{8}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 243, normalized size = 1.31 \begin {gather*} \frac {\frac {12 \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {2 \, {\left (45 \, a^{4} b^{2} \tan \left (d x + c\right )^{2} + 54 \, a^{2} b^{4} \tan \left (d x + c\right )^{2} + 9 \, b^{6} \tan \left (d x + c\right )^{2} + 78 \, a^{5} b \tan \left (d x + c\right ) + 84 \, a^{3} b^{3} \tan \left (d x + c\right ) + 6 \, a b^{5} \tan \left (d x + c\right ) + 34 \, a^{6} + 33 \, a^{4} b^{2} + b^{6}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{7}} + \frac {b^{9} \tan \left (d x + c\right )^{4} - 4 \, a b^{8} \tan \left (d x + c\right )^{3} + 12 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 6 \, b^{9} \tan \left (d x + c\right )^{2} - 40 \, a^{3} b^{6} \tan \left (d x + c\right ) - 36 \, a b^{8} \tan \left (d x + c\right )}{b^{12}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 234, normalized size = 1.26 \begin {gather*} \frac {\frac {11\,a^6+21\,a^4\,b^2+9\,a^2\,b^4-b^6}{2\,b}+\mathrm {tan}\left (c+d\,x\right )\,\left (6\,a^5+12\,a^3\,b^2+6\,a\,b^4\right )}{d\,\left (a^2\,b^6+2\,a\,b^7\,\mathrm {tan}\left (c+d\,x\right )+b^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {3}{2\,b^3}+\frac {3\,a^2}{b^5}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,b^3\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {8\,a^3}{b^6}-\frac {3\,a\,\left (\frac {3}{b^3}+\frac {6\,a^2}{b^5}\right )}{b}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{b^4\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (15\,a^4+18\,a^2\,b^2+3\,b^4\right )}{b^7\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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