Integrand size = 23, antiderivative size = 136 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}-\frac {4 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}+\frac {4 b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^3} \]
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Time = 0.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4516, 3395, 31, 3393, 3384, 3380, 3383} \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {4 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {4 b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {4 b \sin (a+b x) \cos (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2} \]
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Rule 31
Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 3395
Rule 4516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos ^2(a+b x)}{(c+d x)^3}-\frac {\sin ^2(a+b x)}{(c+d x)^3}\right ) \, dx \\ & = 3 \int \frac {\cos ^2(a+b x)}{(c+d x)^3} \, dx-\int \frac {\sin ^2(a+b x)}{(c+d x)^3} \, dx \\ & = -\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \int \frac {1}{c+d x} \, dx}{d^2}+\frac {\left (2 b^2\right ) \int \frac {\sin ^2(a+b x)}{c+d x} \, dx}{d^2}+\frac {\left (3 b^2\right ) \int \frac {1}{c+d x} \, dx}{d^2}-\frac {\left (6 b^2\right ) \int \frac {\cos ^2(a+b x)}{c+d x} \, dx}{d^2} \\ & = -\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac {2 b^2 \log (c+d x)}{d^3}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}+\frac {\left (2 b^2\right ) \int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2}-\frac {\left (6 b^2\right ) \int \left (\frac {1}{2 (c+d x)}+\frac {\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2} \\ & = -\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d^2}-\frac {\left (3 b^2\right ) \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d^2} \\ & = -\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac {\left (b^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}-\frac {\left (3 b^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}+\frac {\left (b^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}+\frac {\left (3 b^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2} \\ & = -\frac {3 \cos ^2(a+b x)}{2 d (c+d x)^2}-\frac {4 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac {\sin ^2(a+b x)}{2 d (c+d x)^2}+\frac {4 b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^3} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {8 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )+\frac {d (d+2 d \cos (2 (a+b x))-4 b (c+d x) \sin (2 (a+b x)))}{(c+d x)^2}-8 b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{2 d^3} \]
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Time = 1.80 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {1}{2 d \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {\frac {4 \,\operatorname {Si}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}}{d}\right )-\frac {b^{3}}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}}{b}\) | \(207\) |
risch | \(-\frac {1}{2 d \left (d x +c \right )^{2}}+\frac {2 b^{2} {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{d^{3}}+\frac {2 b^{2} {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{d^{3}}+\frac {\left (-2 b^{2} d^{3} x^{2}-4 b^{2} c \,d^{2} x -2 b^{2} c^{2} d \right ) \cos \left (2 x b +2 a \right )}{2 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}-\frac {i \left (4 i b^{3} d^{3} x^{3}+12 i b^{3} c \,d^{2} x^{2}+12 i b^{3} c^{2} d x +4 i c^{3} b^{3}\right ) \sin \left (2 x b +2 a \right )}{2 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}\) | \(290\) |
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Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {4 \, d^{2} \cos \left (b x + a\right )^{2} + 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 8 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - d^{2}}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Timed out. \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {2 \, {\left (E_{3}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + E_{3}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left (i \, E_{3}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) - i \, E_{3}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (132) = 264\).
Time = 0.34 (sec) , antiderivative size = 704, normalized size of antiderivative = 5.18 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=-\frac {8 \, b^{5} c^{2} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 16 \, {\left (b x + a\right )} b^{4} c d \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) - 16 \, a b^{4} c d \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 8 \, {\left (b x + a\right )}^{2} b^{3} d^{2} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) - 16 \, {\left (b x + a\right )} a b^{3} d^{2} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 8 \, a^{2} b^{3} d^{2} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 8 \, b^{5} c^{2} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 16 \, {\left (b x + a\right )} b^{4} c d \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) - 16 \, a b^{4} c d \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 8 \, {\left (b x + a\right )}^{2} b^{3} d^{2} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) - 16 \, {\left (b x + a\right )} a b^{3} d^{2} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 8 \, a^{2} b^{3} d^{2} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) - 4 \, b^{4} c d \sin \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b x + a\right )} b^{3} d^{2} \sin \left (2 \, b x + 2 \, a\right ) + 4 \, a b^{3} d^{2} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, b^{3} d^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{3} d^{2}}{2 \, {\left (b^{2} c^{2} d^{3} + 2 \, {\left (b x + a\right )} b c d^{4} - 2 \, a b c d^{4} + {\left (b x + a\right )}^{2} d^{5} - 2 \, {\left (b x + a\right )} a d^{5} + a^{2} d^{5}\right )} b} \]
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Timed out. \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3} \,d x \]
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