Optimal. Leaf size=206 \[ -\frac{a^2 b}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{2 a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{2 d \left (a^2+b^2\right )^3}+\frac{b \left (-8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{a x \left (-14 a^2 b^2+a^4+9 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
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Rubi [A] time = 0.396246, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1647, 1629, 635, 203, 260} \[ -\frac{a^2 b}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{2 a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{2 d \left (a^2+b^2\right )^3}+\frac{b \left (-8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{a x \left (-14 a^2 b^2+a^4+9 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 1647
Rule 1629
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(a+x)^3 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^4 b^2 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac{a^3 b^2 \left (3 a^2+7 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b^2 \left (3 a^4-3 a^2 b^2-2 b^4\right ) x^2}{\left (a^2+b^2\right )^3}+\frac{a b^2 \left (a^2-3 b^2\right ) x^3}{\left (a^2+b^2\right )^3}}{(a+x)^3 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac{\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)^3}+\frac{4 a b^2 \left (-a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)^2}-\frac{2 \left (3 a^4 b^2-8 a^2 b^4+b^6\right )}{\left (a^2+b^2\right )^4 (a+x)}+\frac{b^2 \left (-a \left (a^4-14 a^2 b^2+9 b^4\right )+2 \left (3 a^4-8 a^2 b^2+b^4\right ) x\right )}{\left (a^2+b^2\right )^4 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac{b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac{b \operatorname{Subst}\left (\int \frac{-a \left (a^4-14 a^2 b^2+9 b^4\right )+2 \left (3 a^4-8 a^2 b^2+b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac{b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac{\left (b \left (3 a^4-8 a^2 b^2+b^4\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d}+\frac{\left (a b \left (a^4-14 a^2 b^2+9 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac{a \left (a^4-14 a^2 b^2+9 b^4\right ) x}{2 \left (a^2+b^2\right )^4}+\frac{b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 3.95863, size = 316, normalized size = 1.53 \[ -\frac{b \left (\frac{a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \sin (2 (c+d x))}{2 b}+\left (3 a^2-b^2\right ) \left (a^2+b^2\right ) \cos ^2(c+d x)+\frac{a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \tan ^{-1}(\tan (c+d x))}{b}+\frac{a^2 \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac{4 \left (a^5-a b^4\right )}{a+b \tan (c+d x)}+\left (-8 a^2 b^2-\frac{-8 a^3 b^2+a^5+3 a b^4}{\sqrt{-b^2}}+3 a^4+b^4\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )-2 \left (-8 a^2 b^2+3 a^4+b^4\right ) \log (a+b \tan (c+d x))+\left (-8 a^2 b^2+\frac{-8 a^3 b^2+a^5+3 a b^4}{\sqrt{-b^2}}+3 a^4+b^4\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )\right )}{2 d \left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.115, size = 542, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.84901, size = 625, normalized size = 3.03 \begin{align*} \frac{\frac{{\left (a^{5} - 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{2 \,{\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{{\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{8 \, a^{4} b - 4 \, a^{2} b^{3} +{\left (5 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} +{\left (7 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{5} + 7 \, a^{3} b^{2} - 6 \, a b^{4}\right )} \tan \left (d x + c\right )}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.48319, size = 1152, normalized size = 5.59 \begin{align*} \frac{13 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - b^{7} - 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (a^{5} b^{2} - 14 \, a^{3} b^{4} + 9 \, a b^{6}\right )} d x -{\left (a^{6} b + 23 \, a^{4} b^{3} - 21 \, a^{2} b^{5} - 3 \, b^{7} - 2 \,{\left (a^{7} - 15 \, a^{5} b^{2} + 23 \, a^{3} b^{4} - 9 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + b^{7} +{\left (3 \, a^{6} b - 11 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{5} b^{2} - 8 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \,{\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (4 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + 3 \, a b^{6} +{\left (a^{6} b - 14 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \,{\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23699, size = 651, normalized size = 3.16 \begin{align*} \frac{\frac{{\left (a^{5} - 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{{\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{2 \,{\left (3 \, a^{4} b^{2} - 8 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac{3 \, a^{4} b \tan \left (d x + c\right )^{2} - 8 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + b^{5} \tan \left (d x + c\right )^{2} - a^{5} \tan \left (d x + c\right ) + 2 \, a^{3} b^{2} \tan \left (d x + c\right ) + 3 \, a b^{4} \tan \left (d x + c\right ) - 10 \, a^{2} b^{3} + 2 \, b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac{9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} - 24 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 3 \, b^{7} \tan \left (d x + c\right )^{2} + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) - 48 \, a^{3} b^{4} \tan \left (d x + c\right ) + 2 \, a b^{6} \tan \left (d x + c\right ) + 14 \, a^{6} b - 22 \, a^{4} b^{3}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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