Optimal. Leaf size=211 \[ \frac{b^4}{2 a d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{4 a b^3}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{6 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac{3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.667958, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4397, 2837, 12, 1647, 1629} \[ \frac{b^4}{2 a d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{4 a b^3}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{6 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac{3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2837
Rule 12
Rule 1647
Rule 1629
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^4}{a^4 (b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{a d}\\ &=-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^4 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac{a^4 b^3 \left (7 a^2-3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac{a^2 b^2 \left (3 a^4-9 a^2 b^2+2 b^4\right ) x^2}{\left (a^2-b^2\right )^3}-\frac{a^4 b \left (3 a^2+b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(b+x)^3 \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{2 a^3 d}\\ &=-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{3 a^3 b}{2 (a+b)^4 (a-x)}-\frac{3 a^3 b}{2 (a-b)^4 (a+x)}+\frac{2 a^2 b^4}{\left (a^2-b^2\right )^2 (b+x)^3}-\frac{8 a^4 b^3}{\left (a^2-b^2\right )^3 (b+x)^2}+\frac{12 a^4 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 a^3 d}\\ &=\frac{b^4}{2 a \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac{4 a b^3}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{3 b \log (1-\cos (c+d x))}{4 (a+b)^4 d}+\frac{3 b \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac{6 a b^2 \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}
Mathematica [A] time = 5.47795, size = 184, normalized size = 0.87 \[ \frac{-\frac{48 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}+\frac{4 b^4}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{32 a b^3}{(b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{(a+b)^3}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{(b-a)^3}-\frac{12 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac{12 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{(a-b)^4}}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 220, normalized size = 1. \begin{align*} -{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{3\,b\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{4\, \left ( a-b \right ) ^{4}d}}+{\frac{1}{4\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{4\,d \left ( a+b \right ) ^{4}}}+{\frac{{b}^{4}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}a \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{a{b}^{3}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-6\,{\frac{{a}^{3}{b}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-6\,{\frac{a{b}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.30881, size = 801, normalized size = 3.8 \begin{align*} -\frac{\frac{48 \,{\left (a^{3} b^{2} + a b^{4}\right )} \log \left (a + b - \frac{{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac{12 \, b \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac{2 \,{\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 32 \, a^{3} b^{3} - 37 \, a^{2} b^{4} - 4 \, a b^{5} - 9 \, b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{{\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 84 \, a^{3} b^{3} + 63 \, a^{2} b^{4} - 6 \, a b^{5} + 17 \, b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac{{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{2 \,{\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.10824, size = 2115, normalized size = 10.02 \begin{align*} \text{result too large to display} \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 \frac{\cos{\left (c + d x \right )}}{\left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3658, size = 932, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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