Optimal. Leaf size=232 \[ \frac{a^3}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{a^2 \left (a^2+3 b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{a \left (8 a^2 b^2+a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}+\frac{\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(2 a-b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.482381, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2721, 1647, 1629} \[ \frac{a^3}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{a^2 \left (a^2+3 b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{a \left (8 a^2 b^2+a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}+\frac{\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(2 a-b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 2721
Rule 1647
Rule 1629
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+x)^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a^3 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \left (2 a^4-3 a^2 b^2-3 b^4\right ) x}{\left (a^2-b^2\right )^3}-\frac{a b^4 \left (7 a^2-3 b^2\right ) x^2}{\left (a^2-b^2\right )^3}-\frac{b^4 \left (3 a^2+b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(a+x)^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 b^2 d}\\ &=\frac{\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{b^2 (-2 a+b)}{2 (a+b)^4 (b-x)}-\frac{2 a^3 b^2}{\left (a^2-b^2\right )^2 (a+x)^3}-\frac{2 a^2 b^2 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3 (a+x)^2}-\frac{2 a b^2 \left (a^4+8 a^2 b^2+3 b^4\right )}{\left (a^2-b^2\right )^4 (a+x)}+\frac{b^2 (2 a+b)}{2 (a-b)^4 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 b^2 d}\\ &=\frac{(2 a-b) \log (1-\sin (c+d x))}{4 (a+b)^4 d}+\frac{(2 a+b) \log (1+\sin (c+d x))}{4 (a-b)^4 d}-\frac{a \left (a^4+8 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}+\frac{a^3}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{a^2 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 2.189, size = 196, normalized size = 0.84 \[ \frac{\frac{2 a^3}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{4 a^2 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{4 a \left (8 a^2 b^2+a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}-\frac{1}{(a+b)^3 (\sin (c+d x)-1)}+\frac{1}{(a-b)^3 (\sin (c+d x)+1)}+\frac{(2 a-b) \log (1-\sin (c+d x))}{(a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{(a-b)^4}}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 323, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{5}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-8\,{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{2}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-3\,{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{4}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{{a}^{4}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+3\,{\frac{{a}^{2}{b}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{1}{4\,d \left ( a+b \right ) ^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) a}{2\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) b}{4\,d \left ( a+b \right ) ^{4}}}+{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) a}{2\,d \left ( a-b \right ) ^{4}}}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) b}{4\,d \left ( a-b \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87626, size = 595, normalized size = 2.56 \begin{align*} -\frac{\frac{4 \,{\left (a^{5} + 8 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (b \sin \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 (2 \, a + b\right )} \log \left (\sin \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{{\left (2 \, a - b\right )} \log \left (\sin \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{2 \,{\left (4 \, a^{5} + 8 \, a^{3} b^{2} -{\left (2 \, a^{4} b + 9 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{3} -{\left (3 \, a^{5} + 10 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{2} +{\left (a^{4} b + 11 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\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 )} \sin \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 )} \sin \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 )} \sin \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 )} \sin \left (d x + c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.96383, size = 1727, normalized size = 7.44 \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{\tan ^{3}{\left (c + d x \right )}}{\left (a + b \sin{\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 = 2.10026, size = 626, normalized size = 2.7 \begin{align*} -\frac{\frac{4 \,{\left (a^{5} b + 8 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \log \left ({\left | b \sin \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{{\left (2 \, a + b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac{{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{2 \,{\left (a^{5} \sin \left (d x + c\right )^{2} + 8 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} + 3 \, a b^{4} \sin \left (d x + c\right )^{2} - 3 \, a^{4} b \sin \left (d x + c\right ) + 2 \, a^{2} b^{3} \sin \left (d x + c\right ) + b^{5} \sin \left (d x + c\right ) - 6 \, a^{3} b^{2} - 6 \, a b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\sin \left (d x + c\right )^{2} - 1\right )}} - \frac{2 \,{\left (3 \, a^{5} b^{2} \sin \left (d x + c\right )^{2} + 24 \, a^{3} b^{4} \sin \left (d x + c\right )^{2} + 9 \, a b^{6} \sin \left (d x + c\right )^{2} + 8 \, a^{6} b \sin \left (d x + c\right ) + 52 \, a^{4} b^{3} \sin \left (d x + c\right ) + 12 \, a^{2} b^{5} \sin \left (d x + c\right ) + 6 \, a^{7} + 26 \, a^{5} b^{2} + 4 \, a^{3} b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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