Optimal. Leaf size=212 \[ \frac{3 a^2 \left (a^2+3 b^2\right )}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{3 a^2 b}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{6 a^2 b \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{3 a \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{3 a \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.363985, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4397, 2837, 12, 823, 801} \[ \frac{3 a^2 \left (a^2+3 b^2\right )}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{3 a^2 b}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{6 a^2 b \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{3 a \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{3 a \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2837
Rule 12
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{x}{a (b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{x}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac{(b-a \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{\operatorname{Subst}\left (\int \frac{-3 a^2 b+3 a^2 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{(b-a \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{\operatorname{Subst}\left (\int \left (\frac{3 a (a-b)}{2 (a+b)^3 (a-x)}+\frac{3 a (a+b)}{2 (a-b)^3 (a+x)}-\frac{6 a^2 b}{\left (a^2-b^2\right ) (b+x)^3}+\frac{3 a^2 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^2 (b+x)^2}-\frac{12 a^2 b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac{3 a^2 b}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac{3 a^2 \left (a^2+3 b^2\right )}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac{(b-a \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{3 a \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac{3 a \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac{6 a^2 b \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 = 6.27284, size = 217, normalized size = 1.02 \[ -\frac{a^2 \left (a^2+3 b^2\right )}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}+\frac{6 \left (a^2 b^3+a^4 b\right ) \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4}-\frac{a^2 b}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (a+b)^3}-\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (b-a)^3}+\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d (a+b)^4}-\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d (b-a)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 251, normalized size = 1.2 \begin{align*}{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{3\,a\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 ) a}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{{a}^{2}b}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+3\,{\frac{{a}^{2}{b}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+6\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{4}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+6\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{2}}{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 = 1.26719, size = 805, normalized size = 3.8 \begin{align*} \frac{\frac{48 \,{\left (a^{4} b + a^{2} b^{3}\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 \, a \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 (9 \, a^{6} + 4 \, a^{5} b + 37 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 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 (17 \, a^{6} - 6 \, a^{5} b + 63 \, a^{4} b^{2} - 84 \, 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 = 1.11246, size = 2021, normalized size = 9.53 \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 ^{2}{\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 = 1.39573, size = 930, normalized size = 4.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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