Optimal. Leaf size=74 \[ \frac {a \tan ^4(c+d x)}{4 d}+\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.13, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2607, 30, 2611, 3770} \[ \frac {a \tan ^4(c+d x)}{4 d}+\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2834
Rule 3770
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sin (c+d x)) \tan ^3(c+d x) \, dx &=a \int \sec ^2(c+d x) \tan ^3(c+d x) \, dx+b \int \sec (c+d x) \tan ^4(c+d x) \, dx\\ &=\frac {b \sec (c+d x) \tan ^3(c+d x)}{4 d}-\frac {1}{4} (3 b) \int \sec (c+d x) \tan ^2(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int x^3 \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {1}{8} (3 b) \int \sec (c+d x) \, dx\\ &=\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {a \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 84, normalized size = 1.14 \[ \frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^3(c+d x) \sec (c+d x)}{d}-\frac {b \left (6 \tan (c+d x) \sec ^3(c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 93, normalized size = 1.26 \[ \frac {3 \, b \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, b \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 8 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (5 \, b \cos \left (d x + c\right )^{2} - 2 \, b\right )} \sin \left (d x + c\right ) + 4 \, a}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 81, normalized size = 1.09 \[ \frac {3 \, b \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, b \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (5 \, b \sin \left (d x + c\right )^{3} + 4 \, a \sin \left (d x + c\right )^{2} - 3 \, b \sin \left (d x + c\right ) - 2 \, a\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 114, normalized size = 1.54 \[ \frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {b \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {b \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {b \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {3 b \sin \left (d x +c \right )}{8 d}+\frac {3 b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 89, normalized size = 1.20 \[ \frac {3 \, b \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, b \log \left (\sin \left (d x + c\right ) - 1\right ) + \frac {2 \, {\left (5 \, b \sin \left (d x + c\right )^{3} + 4 \, a \sin \left (d x + c\right )^{2} - 3 \, b \sin \left (d x + c\right ) - 2 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.08, size = 144, normalized size = 1.95 \[ \frac {-\frac {3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {11\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {11\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {3\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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