Optimal. Leaf size=77 \[ \frac {(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac {2 b (a \cos (c+d x)+b)^5}{5 a^3 d}-\frac {\left (a^2-b^2\right ) (a \cos (c+d x)+b)^4}{4 a^3 d} \]
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Rubi [A] time = 0.19, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {4397, 2668, 697} \[ -\frac {\left (a^2-b^2\right ) (a \cos (c+d x)+b)^4}{4 a^3 d}+\frac {(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac {2 b (a \cos (c+d x)+b)^5}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rule 4397
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \sin ^3(c+d x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int (b+x)^3 \left (a^2-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right ) (b+x)^3+2 b (b+x)^4-(b+x)^5\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\left (a^2-b^2\right ) (b+a \cos (c+d x))^4}{4 a^3 d}-\frac {2 b (b+a \cos (c+d x))^5}{5 a^3 d}+\frac {(b+a \cos (c+d x))^6}{6 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 114, normalized size = 1.48 \[ \frac {-45 \left (a^3+8 a b^2\right ) \cos (2 (c+d x))+5 a^3 \cos (6 (c+d x))-360 b \left (a^2+2 b^2\right ) \cos (c+d x)-60 a^2 b \cos (3 (c+d x))+36 a^2 b \cos (5 (c+d x))+90 a b^2 \cos (4 (c+d x))+80 b^3 \cos (3 (c+d x))}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 100, normalized size = 1.30 \[ \frac {10 \, a^{3} \cos \left (d x + c\right )^{6} + 36 \, a^{2} b \cos \left (d x + c\right )^{5} - 90 \, a b^{2} \cos \left (d x + c\right )^{2} - 15 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - 60 \, b^{3} \cos \left (d x + c\right ) - 20 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.11, size = 109, normalized size = 1.42 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+\frac {3 a \,b^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {b^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 95, normalized size = 1.23 \[ \frac {45 \, a b^{2} \sin \left (d x + c\right )^{4} - 5 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} + 12 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b + 20 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} b^{3}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 149, normalized size = 1.94 \[ \frac {32\,a^3}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {4\,{\left (a-b\right )}^3}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {32\,a^2\,\left (5\,a-3\,b\right )}{5\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5}-\frac {8\,{\left (a-b\right )}^2\,\left (7\,a-b\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {12\,a\,\left (3\,a^2-4\,a\,b+b^2\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \cos ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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