Optimal. Leaf size=60 \[ \frac {b \sec ^6(c+d x)}{6 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3567, 3852}
\begin {gather*} \frac {a \tan ^5(c+d x)}{5 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^6(c+d x)}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3852
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \sec ^6(c+d x)}{6 d}+a \int \sec ^6(c+d x) \, dx\\ &=\frac {b \sec ^6(c+d x)}{6 d}-\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {b \sec ^6(c+d x)}{6 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 53, normalized size = 0.88 \begin {gather*} \frac {b \sec ^6(c+d x)}{6 d}+\frac {a \left (\tan (c+d x)+\frac {2}{3} \tan ^3(c+d x)+\frac {1}{5} \tan ^5(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 48, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {b}{6 \cos \left (d x +c \right )^{6}}}{d}\) | \(48\) |
default | \(\frac {-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {b}{6 \cos \left (d x +c \right )^{6}}}{d}\) | \(48\) |
risch | \(\frac {\frac {32 i a \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {32 b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+16 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {32 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{5}+\frac {16 i a}{15}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 70, normalized size = 1.17 \begin {gather*} \frac {5 \, b \tan \left (d x + c\right )^{6} + 6 \, a \tan \left (d x + c\right )^{5} + 15 \, b \tan \left (d x + c\right )^{4} + 20 \, a \tan \left (d x + c\right )^{3} + 15 \, b \tan \left (d x + c\right )^{2} + 30 \, a \tan \left (d x + c\right )}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 57, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (8 \, a \cos \left (d x + c\right )^{5} + 4 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 5 \, b}{30 \, d \cos \left (d x + c\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.58, size = 56, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {a \left (\frac {\tan ^{5}{\left (c + d x \right )}}{5} + \frac {2 \tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right ) + \frac {b \sec ^{6}{\left (c + d x \right )}}{6}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \sec ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 70, normalized size = 1.17 \begin {gather*} \frac {5 \, b \tan \left (d x + c\right )^{6} + 6 \, a \tan \left (d x + c\right )^{5} + 15 \, b \tan \left (d x + c\right )^{4} + 20 \, a \tan \left (d x + c\right )^{3} + 15 \, b \tan \left (d x + c\right )^{2} + 30 \, a \tan \left (d x + c\right )}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.67, size = 68, normalized size = 1.13 \begin {gather*} \frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^6}{6}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{2}+\frac {2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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