Optimal. Leaf size=52 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b \sec ^3(c+d x)}{3 d}+\frac {a \sec (c+d x) \tan (c+d x)}{2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3169, 3853,
3855, 2686, 30} \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x) \sec (c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 3169
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int \left (a \sec ^3(c+d x)+b \sec ^3(c+d x) \tan (c+d x)\right ) \, dx\\ &=a \int \sec ^3(c+d x) \, dx+b \int \sec ^3(c+d x) \tan (c+d x) \, dx\\ &=\frac {a \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a \int \sec (c+d x) \, dx+\frac {b \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b \sec ^3(c+d x)}{3 d}+\frac {a \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 52, normalized size = 1.00 \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b \sec ^3(c+d x)}{3 d}+\frac {a \sec (c+d x) \tan (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 50, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {b}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(50\) |
default | \(\frac {a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {b}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(50\) |
risch | \(\frac {-3 i a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{i \left (d x +c \right )}}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(97\) |
norman | \(\frac {\frac {a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b}{3 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 61, normalized size = 1.17 \begin {gather*} -\frac {3 \, a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {4 \, b}{\cos \left (d x + c\right )^{3}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.93, size = 74, normalized size = 1.42 \begin {gather*} \frac {3 \, a \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 6 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, b}{12 \, d \cos \left (d x + c\right )^{3}} \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 \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (46) = 92\).
time = 0.43, size = 99, normalized size = 1.90 \begin {gather*} \frac {3 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.21, size = 105, normalized size = 2.02 \begin {gather*} \frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,b}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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