Optimal. Leaf size=116 \[ -\frac {(8 a+15 b) \log (1-\sin (c+d x))}{16 d}-\frac {(8 a-15 b) \log (\sin (c+d x)+1)}{16 d}+\frac {\tan ^4(c+d x) (a+b \sin (c+d x))}{4 d}-\frac {\tan ^2(c+d x) (4 a+5 b \sin (c+d x))}{8 d}-\frac {15 b \sin (c+d x)}{8 d} \]
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Rubi [A] time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2721, 819, 774, 633, 31} \[ -\frac {(8 a+15 b) \log (1-\sin (c+d x))}{16 d}-\frac {(8 a-15 b) \log (\sin (c+d x)+1)}{16 d}+\frac {\tan ^4(c+d x) (a+b \sin (c+d x))}{4 d}-\frac {\tan ^2(c+d x) (4 a+5 b \sin (c+d x))}{8 d}-\frac {15 b \sin (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 774
Rule 819
Rule 2721
Rubi steps
\begin {align*} \int (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5 (a+x)}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {(a+b \sin (c+d x)) \tan ^4(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (4 a b^2+5 b^2 x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b^2 d}\\ &=-\frac {(4 a+5 b \sin (c+d x)) \tan ^2(c+d x)}{8 d}+\frac {(a+b \sin (c+d x)) \tan ^4(c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {x \left (8 a b^4+15 b^4 x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=-\frac {15 b \sin (c+d x)}{8 d}-\frac {(4 a+5 b \sin (c+d x)) \tan ^2(c+d x)}{8 d}+\frac {(a+b \sin (c+d x)) \tan ^4(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {-15 b^6-8 a b^4 x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=-\frac {15 b \sin (c+d x)}{8 d}-\frac {(4 a+5 b \sin (c+d x)) \tan ^2(c+d x)}{8 d}+\frac {(a+b \sin (c+d x)) \tan ^4(c+d x)}{4 d}+\frac {(8 a-15 b) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {(8 a+15 b) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=-\frac {(8 a+15 b) \log (1-\sin (c+d x))}{16 d}-\frac {(8 a-15 b) \log (1+\sin (c+d x))}{16 d}-\frac {15 b \sin (c+d x)}{8 d}-\frac {(4 a+5 b \sin (c+d x)) \tan ^2(c+d x)}{8 d}+\frac {(a+b \sin (c+d x)) \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 123, normalized size = 1.06 \[ -\frac {a \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d}-\frac {b \sin (c+d x) \tan ^4(c+d x)}{d}-\frac {5 b \left (6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 114, normalized size = 0.98 \[ -\frac {{\left (8 \, a - 15 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (8 \, a + 15 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 16 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, b \cos \left (d x + c\right )^{4} + 9 \, b \cos \left (d x + c\right )^{2} - 2 \, b\right )} \sin \left (d x + c\right ) - 4 \, a}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 108, normalized size = 0.93 \[ -\frac {{\left (8 \, a - 15 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (8 \, a + 15 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 16 \, b \sin \left (d x + c\right ) - \frac {2 \, {\left (6 \, a \sin \left (d x + c\right )^{4} + 9 \, b \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} - 7 \, b \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 147, normalized size = 1.27 \[ \frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 b \left (\sin ^{7}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {3 b \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {5 b \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {15 b \sin \left (d x +c \right )}{8 d}+\frac {15 b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 108, normalized size = 0.93 \[ -\frac {{\left (8 \, a - 15 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (8 \, a + 15 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, b \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, b \sin \left (d x + c\right )^{3} + 8 \, a \sin \left (d x + c\right )^{2} - 7 \, b \sin \left (d x + c\right ) - 6 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.10, size = 261, normalized size = 2.25 \[ \frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (a-\frac {15\,b}{8}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (a+\frac {15\,b}{8}\right )}{d}-\frac {\frac {15\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-10\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {9\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-10\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {15\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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