Optimal. Leaf size=120 \[ \frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {b \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac {3 a^2 b \cos ^4(c+d x)}{4 d}-\frac {3 a b^2 \cos (c+d x)}{d}-\frac {b^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.19, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4397, 2837, 12, 894} \[ -\frac {a \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {b \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac {3 a^2 b \cos ^4(c+d x)}{4 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a b^2 \cos (c+d x)}{d}-\frac {b^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rule 4397
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \sin ^2(c+d x) \tan (c+d x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a (b+x)^3 \left (a^2-x^2\right )}{x} \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(b+x)^3 \left (a^2-x^2\right )}{x} \, dx,x,a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (3 a^2 b^2+\frac {a^2 b^3}{x}+b \left (3 a^2-b^2\right ) x+\left (a^2-3 b^2\right ) x^2-3 b x^3-x^4\right ) \, dx,x,a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {3 a b^2 \cos (c+d x)}{d}-\frac {b \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}-\frac {a \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}+\frac {3 a^2 b \cos ^4(c+d x)}{4 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {b^3 \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 106, normalized size = 0.88 \[ -\frac {-\frac {1}{5} a^3 \cos ^5(c+d x)+\frac {1}{3} a \left (a^2-3 b^2\right ) \cos ^3(c+d x)+\frac {1}{2} b \left (3 a^2-b^2\right ) \cos ^2(c+d x)-\frac {3}{4} a^2 b \cos ^4(c+d x)+3 a b^2 \cos (c+d x)+b^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 101, normalized size = 0.84 \[ \frac {12 \, a^{3} \cos \left (d x + c\right )^{5} + 45 \, a^{2} b \cos \left (d x + c\right )^{4} - 180 \, a b^{2} \cos \left (d x + c\right ) - 20 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 60 \, b^{3} \log \left (-\cos \left (d x + c\right )\right ) - 30 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}}{60 \, d} \]
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 = 128, normalized size = 1.07 \[ -\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5 d}-\frac {2 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d}+\frac {3 a^{2} b \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {\cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a \,b^{2}}{d}-\frac {2 a \,b^{2} \cos \left (d x +c \right )}{d}-\frac {b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 94, normalized size = 0.78 \[ \frac {45 \, a^{2} b \sin \left (d x + c\right )^{4} + 4 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{3} + 60 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a b^{2} - 30 \, {\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} b^{3}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 237, normalized size = 1.98 \[ \frac {40\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3\,d}-\frac {4\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {16\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}+\frac {32\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5\,d}-\frac {2\,b^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {2\,b^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}+\frac {2\,b^3\,\mathrm {atanh}\left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}-1\right )}{d}-\frac {12\,a\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}+\frac {12\,a^2\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}+\frac {8\,a\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d}-\frac {24\,a^2\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d}+\frac {12\,a^2\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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