Optimal. Leaf size=232 \[ \frac {a^2 \left (a^2+3 b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac {a \left (a^4+8 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}+\frac {a^3}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {(2 a-b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac {(2 a+b) \log (\sin (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.48, antiderivative size = 232, normalized size of antiderivative = 1.00, 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 1629
Rule 1647
Rule 2721
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*}
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Mathematica [A] time = 2.28, size = 196, normalized size = 0.84 \[ \frac {\frac {4 a^2 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {4 a \left (a^4+8 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac {2 a^3}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {1}{(a+b)^3 (\sin (c+d x)-1)}+\frac {1}{(a-b)^3 (\sin (c+d x)+1)}+\frac {(2 a-b) \log (1-\sin (c+d x))}{(a+b)^4}+\frac {(2 a+b) \log (\sin (c+d x)+1)}{(a-b)^4}}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 788, normalized size = 3.40 \[ -\frac {2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} + 2 \, {\left (3 \, a^{7} + 7 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{5} b^{2} + 8 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 8 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{7} + 9 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left ({\left (2 \, a^{5} b^{2} + 9 \, a^{4} b^{3} + 16 \, a^{3} b^{4} + 14 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, a^{6} b + 9 \, a^{5} b^{2} + 16 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (2 \, a^{7} + 9 \, a^{6} b + 18 \, a^{5} b^{2} + 23 \, a^{4} b^{3} + 22 \, a^{3} b^{4} + 15 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (2 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 16 \, a^{3} b^{4} - 14 \, a^{2} b^{5} + 6 \, a b^{6} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, a^{6} b - 9 \, a^{5} b^{2} + 16 \, a^{4} b^{3} - 14 \, a^{3} b^{4} + 6 \, a^{2} b^{5} - a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (2 \, a^{7} - 9 \, a^{6} b + 18 \, a^{5} b^{2} - 23 \, a^{4} b^{3} + 22 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left (2 \, a^{6} b + 7 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} - 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 )^{2} \sin \left (d x + c\right ) - {\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}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.39, size = 464, normalized size = 2.00 \[ -\frac {\frac {4 \, {\left (a^{5} b + 8 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} - \frac {{\left (2 \, a + b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (a^{5} \sin \left (d x + c\right )^{2} + 8 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} + 3 \, a b^{4} \sin \left (d x + c\right )^{2} - 3 \, a^{4} b \sin \left (d x + c\right ) + 2 \, a^{2} b^{3} \sin \left (d x + c\right ) + b^{5} \sin \left (d x + c\right ) - 6 \, a^{3} b^{2} - 6 \, a b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}} - \frac {2 \, {\left (3 \, a^{5} b^{2} \sin \left (d x + c\right )^{2} + 24 \, a^{3} b^{4} \sin \left (d x + c\right )^{2} + 9 \, a b^{6} \sin \left (d x + c\right )^{2} + 8 \, a^{6} b \sin \left (d x + c\right ) + 52 \, a^{4} b^{3} \sin \left (d x + c\right ) + 12 \, a^{2} b^{5} \sin \left (d x + c\right ) + 6 \, a^{7} + 26 \, a^{5} b^{2} + 4 \, a^{3} b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 323, normalized size = 1.39 \[ -\frac {1}{4 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\ln \left (\sin \left (d x +c \right )-1\right ) a}{2 d \left (a +b \right )^{4}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) b}{4 d \left (a +b \right )^{4}}+\frac {a^{3}}{2 d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {8 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right ) b^{2}}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {3 a \ln \left (a +b \sin \left (d x +c \right )\right ) b^{4}}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {a^{4}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {3 a^{2} b^{2}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{4 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) a}{2 d \left (a -b \right )^{4}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) b}{4 d \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 441, normalized size = 1.90 \[ -\frac {\frac {4 \, {\left (a^{5} + 8 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (2 \, a + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (2 \, a - b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (4 \, a^{5} + 8 \, a^{3} b^{2} - {\left (2 \, a^{4} b + 9 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{3} - {\left (3 \, a^{5} + 10 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{2} + {\left (a^{4} b + 11 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )}}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.46, size = 690, normalized size = 2.97 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (2\,a-b\right )}{2\,d\,{\left (a+b\right )}^4}-\frac {\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-a^5+6\,a^3\,b^2+7\,a\,b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^5+2\,a^3\,b^2+9\,a\,b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^4\,b+13\,a^2\,b^3-4\,b^5\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-a^5+6\,a^3\,b^2+7\,a\,b^4\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (7\,a^4+5\,a^2\,b^2\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^4+13\,a^2\,b^2-4\,b^4\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a^3\,b+5\,a\,b^3\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^2+8\,b^2\right )+4\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^2-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (2\,a+b\right )}{2\,d\,{\left (a-b\right )}^4}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^5+8\,a^3\,b^2+3\,a\,b^4\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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