Optimal. Leaf size=120 \[ -\frac{a \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac{b \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac{3 a^2 b \cos ^4(c+d x)}{4 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a b^2 \cos (c+d x)}{d}-\frac{b^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.194266, antiderivative size = 120, normalized size of antiderivative = 1., 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 4397
Rule 2837
Rule 12
Rule 894
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*}
Mathematica [A] time = 0.185859, size = 106, normalized size = 0.88 \[ -\frac{\frac{1}{3} a \left (a^2-3 b^2\right ) \cos ^3(c+d x)+\frac{1}{2} b \left (3 a^2-b^2\right ) \cos ^2(c+d x)-\frac{3}{4} a^2 b \cos ^4(c+d x)-\frac{1}{5} a^3 \cos ^5(c+d x)+3 a b^2 \cos (c+d x)+b^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 128, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{5\,d}}-{\frac{2\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15\,d}}+{\frac{3\,{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}a{b}^{2}}{d}}-2\,{\frac{a{b}^{2}\cos \left ( dx+c \right ) }{d}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16498, size = 127, normalized size = 1.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.530426, size = 247, normalized size = 2.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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