Optimal. Leaf size=61 \[ \frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x))}{4 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2747, 653, 205,
212} \begin {gather*} \frac {\sec ^4(c+d x) (a \sin (c+d x)+b)}{4 d}+\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 653
Rule 2747
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b^5 \text {Subst}\left (\int \frac {a+x}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^4(c+d x) (b+a \sin (c+d x))}{4 d}+\frac {\left (3 a b^3\right ) \text {Subst}\left (\int \frac {1}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac {\sec ^4(c+d x) (b+a \sin (c+d x))}{4 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(3 a b) \text {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x))}{4 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 68, normalized size = 1.11 \begin {gather*} \frac {b \sec ^4(c+d x)}{4 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {3 a \left (\tanh ^{-1}(\sin (c+d x))+\sec (c+d x) \tan (c+d x)\right )}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 63, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(63\) |
default | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(63\) |
risch | \(\frac {-3 i a \,{\mathrm e}^{7 i \left (d x +c \right )}-11 i a \,{\mathrm e}^{5 i \left (d x +c \right )}+11 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+16 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{i \left (d x +c \right )}}{4 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{4}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(123\) |
norman | \(\frac {\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {2 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 78, normalized size = 1.28 \begin {gather*} \frac {3 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a \sin \left (d x + c\right )^{3} - 5 \, a \sin \left (d x + c\right ) - 2 \, b\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 82, normalized size = 1.34 \begin {gather*} \frac {3 \, a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right ) + 4 \, b}{16 \, d \cos \left (d x + c\right )^{4}} \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 \sin {\left (c + d x \right )}\right ) \sec ^{5}{\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 = 6.63, size = 70, normalized size = 1.15 \begin {gather*} \frac {3 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a \sin \left (d x + c\right )^{3} - 5 \, a \sin \left (d x + c\right ) - 2 \, b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.14, size = 64, normalized size = 1.05 \begin {gather*} \frac {3\,a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{8\,d}+\frac {-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{8}+\frac {5\,a\,\sin \left (c+d\,x\right )}{8}+\frac {b}{4}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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