Optimal. Leaf size=168 \[ \frac{b \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}+\frac{a^2 b \sin (c+d x)}{d \left (a^2+b^2\right )^2}+\frac{a \cos ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac{a \cos (c+d x)}{d \left (a^2+b^2\right )}+\frac{a b^2 \cos (c+d x)}{d \left (a^2+b^2\right )^2}+\frac{a^3 b \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{5/2}} \]
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Rubi [A] time = 0.22299, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3518, 3109, 2564, 30, 2633, 3099, 3074, 206, 2638} \[ \frac{b \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}+\frac{a^2 b \sin (c+d x)}{d \left (a^2+b^2\right )^2}+\frac{a \cos ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac{a \cos (c+d x)}{d \left (a^2+b^2\right )}+\frac{a b^2 \cos (c+d x)}{d \left (a^2+b^2\right )^2}+\frac{a^3 b \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3518
Rule 3109
Rule 2564
Rule 30
Rule 2633
Rule 3099
Rule 3074
Rule 206
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx &=\int \frac{\cos (c+d x) \sin ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\\ &=\frac{a \int \sin ^3(c+d x) \, dx}{a^2+b^2}+\frac{b \int \cos (c+d x) \sin ^2(c+d x) \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\sin ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2}\\ &=\frac{a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac{\left (a^3 b\right ) \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \sin (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac{a \cos (c+d x)}{\left (a^2+b^2\right ) d}+\frac{a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac{a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}+\frac{b \sin ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac{\left (a^3 b\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac{a^3 b \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}+\frac{a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac{a \cos (c+d x)}{\left (a^2+b^2\right ) d}+\frac{a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac{a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}+\frac{b \sin ^3(c+d x)}{3 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 1.37492, size = 139, normalized size = 0.83 \[ \frac{\sqrt{a^2+b^2} \left (\left (3 a b^2-9 a^3\right ) \cos (c+d x)+a \left (a^2+b^2\right ) \cos (3 (c+d x))-2 b \sin (c+d x) \left (\left (a^2+b^2\right ) \cos (2 (c+d x))-7 a^2-b^2\right )\right )-24 a^3 b \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{12 d \left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 205, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{-b{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-a{b}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( -10/3\,b{a}^{2}-4/3\,{b}^{3} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+2\,{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-b{a}^{2}\tan \left ( 1/2\,dx+c/2 \right ) +2/3\,{a}^{3}-1/3\,a{b}^{2}}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-16\,{\frac{b{a}^{3}}{ \left ( 8\,{a}^{4}+16\,{a}^{2}{b}^{2}+8\,{b}^{4} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27774, size = 597, normalized size = 3.55 \begin{align*} \frac{3 \, \sqrt{a^{2} + b^{2}} a^{3} b \log \left (\frac{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} - 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{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 (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - 6 \,{\left (a^{5} + a^{3} b^{2}\right )} \cos \left (d x + c\right ) + 2 \,{\left (4 \, a^{4} b + 5 \, a^{2} b^{3} + b^{5} -{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.38606, size = 325, normalized size = 1.93 \begin{align*} \frac{\frac{3 \, a^{3} b \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (3 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 10 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a^{3} + a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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