Optimal. Leaf size=228 \[ -\frac{a b \left (11 a^2+b^2\right )}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{a \left (2 a^2+b^2\right )}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{2 a^3 \left (a^2+5 b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) (a-b \cos (c+d x))}{2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{(4 a+b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(4 a-b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.40793, antiderivative size = 228, 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, 2668, 741, 801} \[ -\frac{a b \left (11 a^2+b^2\right )}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{a \left (2 a^2+b^2\right )}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{2 a^3 \left (a^2+5 b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) (a-b \cos (c+d x))}{2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{(4 a+b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(4 a-b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2668
Rule 741
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac{\csc ^3(c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{(a-b \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac{a \operatorname{Subst}\left (\int \frac{-4 a^2+b^2+3 b x}{(b+x)^3 \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac{(a-b \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac{a \operatorname{Subst}\left (\int \left (\frac{(-4 a-b) (a-b)}{2 a (a+b)^3 (a-x)}+\frac{(4 a-b) (a+b)}{2 a (a-b)^3 (a+x)}-\frac{2 \left (2 a^2+b^2\right )}{(a-b) (a+b) (b+x)^3}+\frac{b \left (11 a^2+b^2\right )}{(a-b)^2 (a+b)^2 (b+x)^2}-\frac{4 \left (a^4+5 a^2 b^2\right )}{(a-b)^3 (a+b)^3 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=\frac{a \left (2 a^2+b^2\right )}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac{a b \left (11 a^2+b^2\right )}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac{(a-b \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac{(4 a+b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}+\frac{(4 a-b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac{2 a^3 \left (a^2+5 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}
Mathematica [A] time = 6.26157, size = 217, normalized size = 0.95 \[ -\frac{2 \left (5 a^3 b^2+a^5\right ) \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4}+\frac{4 a^3 b}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}+\frac{a^3}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (a+b)^3}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (b-a)^3}+\frac{(4 a+b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d (a+b)^4}+\frac{(4 a-b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d (b-a)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 256, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{a\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{ \left ( a-b \right ) ^{4}d}}-{\frac{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{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{d \left ( a+b \right ) ^{4}}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{4\,d \left ( a+b \right ) ^{4}}}+{\frac{{a}^{3}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{{a}^{3}b}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-2\,{\frac{{a}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-10\,{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23796, size = 798, normalized size = 3.5 \begin{align*} -\frac{\frac{16 \,{\left (a^{5} + 5 \, a^{3} b^{2}\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{4 \,{\left (4 \, a + b\right )} \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} - 44 \, a^{5} b - 35 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{{\left (15 \, a^{6} + 70 \, a^{5} b - 95 \, a^{4} b^{2} + 20 \, a^{3} b^{3} - 15 \, a^{2} b^{4} + 6 \, a b^{5} - 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 = 1.16053, size = 2142, normalized size = 9.39 \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{\sec ^{3}{\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 = 1.39256, size = 911, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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