Optimal. Leaf size=72 \[ a x-\frac {b \cos (c+d x)}{d}-\frac {2 b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2801, 3554, 8,
2670, 276} \begin {gather*} \frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x-\frac {b \cos (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {2 b \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 276
Rule 2670
Rule 2801
Rule 3554
Rubi steps
\begin {align*} \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx &=\int \left (a \tan ^4(c+d x)+b \sin (c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a \int \tan ^4(c+d x) \, dx+b \int \sin (c+d x) \tan ^4(c+d x) \, dx\\ &=\frac {a \tan ^3(c+d x)}{3 d}-a \int \tan ^2(c+d x) \, dx-\frac {b \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}+a \int 1 \, dx-\frac {b \text {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a x-\frac {b \cos (c+d x)}{d}-\frac {2 b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 81, normalized size = 1.12 \begin {gather*} \frac {a \tan ^{-1}(\tan (c+d x))}{d}-\frac {b \cos (c+d x)}{d}-\frac {2 b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 98, normalized size = 1.36
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(98\) |
default | \(\frac {a \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(98\) |
risch | \(a x -\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {4 \left (3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a +3 b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{3}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 65, normalized size = 0.90 \begin {gather*} \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a - b {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 73, normalized size = 1.01 \begin {gather*} \frac {3 \, a d x \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )^{4} - 6 \, b \cos \left (d x + c\right )^{2} - {\left (4 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) + b}{3 \, 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 + b \sin {\left (c + d x \right )}\right ) \tan ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.18, size = 110, normalized size = 1.53 \begin {gather*} a\,x+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {32\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {16\,b}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^3\,\left ({\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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