Integrand size = 23, antiderivative size = 205 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=-\frac {2 b^2}{3 d^3 (c+d x)}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {8 b^3 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {8 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \]
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Time = 0.45 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4516, 3395, 32, 3394, 12, 3384, 3380, 3383} \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=\frac {8 b^3 \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {8 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {4 b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}-\frac {2 b^2}{3 d^3 (c+d x)} \]
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Rule 12
Rule 32
Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 3395
Rule 4516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos ^2(a+b x)}{(c+d x)^4}-\frac {\sin ^2(a+b x)}{(c+d x)^4}\right ) \, dx \\ & = 3 \int \frac {\cos ^2(a+b x)}{(c+d x)^4} \, dx-\int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx \\ & = -\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2 \int \frac {1}{(c+d x)^2} \, dx}{3 d^2}+\frac {\left (2 b^2\right ) \int \frac {\sin ^2(a+b x)}{(c+d x)^2} \, dx}{3 d^2}+\frac {b^2 \int \frac {1}{(c+d x)^2} \, dx}{d^2}-\frac {\left (2 b^2\right ) \int \frac {\cos ^2(a+b x)}{(c+d x)^2} \, dx}{d^2} \\ & = -\frac {2 b^2}{3 d^3 (c+d x)}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {\left (4 b^3\right ) \int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx}{3 d^3}-\frac {\left (4 b^3\right ) \int -\frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx}{d^3} \\ & = -\frac {2 b^2}{3 d^3 (c+d x)}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {\left (2 b^3\right ) \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx}{3 d^3}+\frac {\left (2 b^3\right ) \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx}{d^3} \\ & = -\frac {2 b^2}{3 d^3 (c+d x)}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {\left (2 b^3 \cos \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}{3 d^3}+\frac {\left (2 b^3 \cos \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^3}+\frac {\left (2 b^3 \sin \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}{3 d^3}+\frac {\left (2 b^3 \sin \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^3} \\ & = -\frac {2 b^2}{3 d^3 (c+d x)}-\frac {\cos ^2(a+b x)}{d (c+d x)^3}+\frac {2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac {8 b^3 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}+\frac {4 b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}+\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac {8 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.61 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=\frac {8 b^3 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\frac {d \left (\left (-2 d^2+4 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+d (-d+2 b (c+d x) \sin (2 (a+b x)))\right )}{(c+d x)^3}+8 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{3 d^4} \]
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Time = 3.63 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {1}{3 d \left (d x +c \right )^{3}}+\frac {b^{4} \left (-\frac {2 \cos \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}-\frac {2 \left (-\frac {\sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )-\frac {2 b^{4}}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}}{b}\) | \(243\) |
risch | \(-\frac {1}{3 d \left (d x +c \right )^{3}}-\frac {4 i b^{3} {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{3 d^{4}}+\frac {4 i b^{3} {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}-\frac {\left (-4 b^{5} d^{5} x^{5}-20 b^{5} c \,d^{4} x^{4}-40 b^{5} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{2}-20 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-4 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \cos \left (2 x b +2 a \right )}{3 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}-\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \sin \left (2 x b +2 a \right )}{3 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}\) | \(423\) |
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Time = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.40 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=-\frac {4 \, b^{2} d^{3} x^{2} + 8 \, b^{2} c d^{2} x + 4 \, b^{2} c^{2} d - d^{3} - 4 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} - 4 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{3 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
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Timed out. \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.70 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=-\frac {3 \, {\left (E_{4}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + E_{4}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, {\left (i \, E_{4}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) - i \, E_{4}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 1}{3 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (193) = 386\).
Time = 0.38 (sec) , antiderivative size = 1305, normalized size of antiderivative = 6.37 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^4} \,d x \]
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