Integrand size = 30, antiderivative size = 191 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\frac {-c \cos (d+e x)+b \sin (d+e x)}{5 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{15 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{15 c \left (b^2+c^2\right ) e (c \cos (d+e x)-b \sin (d+e x))} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3195, 3193} \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{15 e \left (b^2+c^2\right ) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{15 c e \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x))} \]
[In]
[Out]
Rule 3193
Rule 3195
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (d+e x)-b \sin (d+e x)}{5 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}+\frac {2 \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2} \, dx}{5 \sqrt {b^2+c^2}} \\ & = -\frac {c \cos (d+e x)-b \sin (d+e x)}{5 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{15 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}+\frac {2 \int \frac {1}{\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx}{15 \left (b^2+c^2\right )} \\ & = -\frac {c \cos (d+e x)-b \sin (d+e x)}{5 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{15 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{15 c \left (b^2+c^2\right ) e (c \cos (d+e x)-b \sin (d+e x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(420\) vs. \(2(191)=382\).
Time = 1.84 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\frac {-76 b^4 c-152 b^2 c^3-76 c^5+90 b c \left (b^2+c^2\right )^{3/2} \cos (d+e x)+20 c \left (-b^4+c^4\right ) \cos (2 (d+e x))+10 b^3 c \sqrt {b^2+c^2} \cos (3 (d+e x))+10 b c^3 \sqrt {b^2+c^2} \cos (3 (d+e x))-4 b^3 c \sqrt {b^2+c^2} \cos (5 (d+e x))+4 b c^3 \sqrt {b^2+c^2} \cos (5 (d+e x))+10 b^4 \sqrt {b^2+c^2} \sin (d+e x)+110 b^2 c^2 \sqrt {b^2+c^2} \sin (d+e x)+100 c^4 \sqrt {b^2+c^2} \sin (d+e x)-40 b^3 c^2 \sin (2 (d+e x))-40 b c^4 \sin (2 (d+e x))-5 b^4 \sqrt {b^2+c^2} \sin (3 (d+e x))+5 c^4 \sqrt {b^2+c^2} \sin (3 (d+e x))+b^4 \sqrt {b^2+c^2} \sin (5 (d+e x))-6 b^2 c^2 \sqrt {b^2+c^2} \sin (5 (d+e x))+c^4 \sqrt {b^2+c^2} \sin (5 (d+e x))}{120 c \left (b^2+c^2\right ) e (c \cos (d+e x)-b \sin (d+e x))^5} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {4 \left (5 i \sqrt {b^{2}+c^{2}}\, b^{2} c \,{\mathrm e}^{i \left (e x +d \right )}+5 i \sqrt {b^{2}+c^{2}}\, c^{3} {\mathrm e}^{i \left (e x +d \right )}+10 b^{4} {\mathrm e}^{2 i \left (e x +d \right )}+20 b^{2} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+10 c^{4} {\mathrm e}^{2 i \left (e x +d \right )}+2 i b^{3} c +2 i b \,c^{3}+5 \sqrt {b^{2}+c^{2}}\, b^{3} {\mathrm e}^{i \left (e x +d \right )}+5 \sqrt {b^{2}+c^{2}}\, b \,c^{2} {\mathrm e}^{i \left (e x +d \right )}+b^{4}-c^{4}\right ) \left (i b^{3}-3 i b \,c^{2}-3 b^{2} c +c^{3}\right )}{15 e \left (i \sqrt {b^{2}+c^{2}}\, c +b^{2} {\mathrm e}^{i \left (e x +d \right )}+c^{2} {\mathrm e}^{i \left (e x +d \right )}+\sqrt {b^{2}+c^{2}}\, b \right )^{5}}\) | \(246\) |
derivativedivides | \(\frac {-\frac {2 \left (4 \sqrt {b^{2}+c^{2}}\, b^{2}+\sqrt {b^{2}+c^{2}}\, c^{2}+4 b^{3}+3 c^{2} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{2}}-\frac {4 \left (8 \sqrt {b^{2}+c^{2}}\, b^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+8 b^{4}+8 b^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{3}}-\frac {8 \left (24 \sqrt {b^{2}+c^{2}}\, b^{4}+20 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+2 \sqrt {b^{2}+c^{2}}\, c^{4}+24 b^{5}+32 b^{3} c^{2}+9 c^{4} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 c^{4}}-\frac {4 \left (48 b^{6}+76 b^{4} c^{2}+31 b^{2} c^{4}+2 c^{6}+48 \sqrt {b^{2}+c^{2}}\, b^{5}+52 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+11 \sqrt {b^{2}+c^{2}}\, b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{3 c^{5}}-\frac {2 \left (192 \sqrt {b^{2}+c^{2}}\, b^{6}+256 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+96 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+7 c^{6} \sqrt {b^{2}+c^{2}}+192 b^{7}+352 b^{5} c^{2}+200 b^{3} c^{4}+35 c^{6} b \right )}{15 c^{6}}}{e \,c^{4} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \sqrt {b^{2}+c^{2}}}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right )^{2} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(496\) |
default | \(\frac {-\frac {2 \left (4 \sqrt {b^{2}+c^{2}}\, b^{2}+\sqrt {b^{2}+c^{2}}\, c^{2}+4 b^{3}+3 c^{2} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{2}}-\frac {4 \left (8 \sqrt {b^{2}+c^{2}}\, b^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+8 b^{4}+8 b^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{3}}-\frac {8 \left (24 \sqrt {b^{2}+c^{2}}\, b^{4}+20 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+2 \sqrt {b^{2}+c^{2}}\, c^{4}+24 b^{5}+32 b^{3} c^{2}+9 c^{4} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 c^{4}}-\frac {4 \left (48 b^{6}+76 b^{4} c^{2}+31 b^{2} c^{4}+2 c^{6}+48 \sqrt {b^{2}+c^{2}}\, b^{5}+52 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+11 \sqrt {b^{2}+c^{2}}\, b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{3 c^{5}}-\frac {2 \left (192 \sqrt {b^{2}+c^{2}}\, b^{6}+256 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+96 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+7 c^{6} \sqrt {b^{2}+c^{2}}+192 b^{7}+352 b^{5} c^{2}+200 b^{3} c^{4}+35 c^{6} b \right )}{15 c^{6}}}{e \,c^{4} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \sqrt {b^{2}+c^{2}}}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right )^{2} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(496\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (178) = 356\).
Time = 0.30 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {7 \, b^{6} + 26 \, b^{4} c^{2} + 31 \, b^{2} c^{4} + 12 \, c^{6} + 5 \, {\left (b^{6} + b^{4} c^{2} - b^{2} c^{4} - c^{6}\right )} \cos \left (e x + d\right )^{2} + 10 \, {\left (b^{5} c + 2 \, b^{3} c^{3} + b c^{5}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - {\left (2 \, {\left (b^{5} - 10 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \cos \left (e x + d\right )^{5} - 5 \, {\left (b^{5} - 6 \, b^{3} c^{2} + b c^{4}\right )} \cos \left (e x + d\right )^{3} + 5 \, {\left (3 \, b^{5} + 3 \, b^{3} c^{2} + 2 \, b c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, b^{4} c + 25 \, b^{2} c^{3} + 12 \, c^{5} + 2 \, {\left (5 \, b^{4} c - 10 \, b^{2} c^{3} + c^{5}\right )} \cos \left (e x + d\right )^{4} - {\left (15 \, b^{4} c - 10 \, b^{2} c^{3} - c^{5}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}}{15 \, {\left ({\left (5 \, b^{8} c - 14 \, b^{4} c^{5} - 8 \, b^{2} c^{7} + c^{9}\right )} e \cos \left (e x + d\right )^{5} - 10 \, {\left (b^{8} c + b^{6} c^{3} - b^{4} c^{5} - b^{2} c^{7}\right )} e \cos \left (e x + d\right )^{3} + 5 \, {\left (b^{8} c + 2 \, b^{6} c^{3} + b^{4} c^{5}\right )} e \cos \left (e x + d\right ) - {\left ({\left (b^{9} - 8 \, b^{7} c^{2} - 14 \, b^{5} c^{4} + 5 \, b c^{8}\right )} e \cos \left (e x + d\right )^{4} - 2 \, {\left (b^{9} - 3 \, b^{7} c^{2} - 9 \, b^{5} c^{4} - 5 \, b^{3} c^{6}\right )} e \cos \left (e x + d\right )^{2} + {\left (b^{9} + 2 \, b^{7} c^{2} + b^{5} c^{4}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {2 \, {\left (192 \, b^{7} + 352 \, b^{5} c^{2} + 200 \, b^{3} c^{4} + 35 \, b c^{6} + 15 \, {\left (4 \, b^{3} c^{4} + 3 \, b c^{6} + {\left (4 \, b^{2} c^{4} + c^{6}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 30 \, {\left (8 \, b^{4} c^{3} + 8 \, b^{2} c^{5} + c^{7} + 4 \, {\left (2 \, b^{3} c^{3} + b c^{5}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 20 \, {\left (24 \, b^{5} c^{2} + 32 \, b^{3} c^{4} + 9 \, b c^{6} + 2 \, {\left (12 \, b^{4} c^{2} + 10 \, b^{2} c^{4} + c^{6}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 10 \, {\left (48 \, b^{6} c + 76 \, b^{4} c^{3} + 31 \, b^{2} c^{5} + 2 \, c^{7} + {\left (48 \, b^{5} c + 52 \, b^{3} c^{3} + 11 \, b c^{5}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + {\left (192 \, b^{6} + 256 \, b^{4} c^{2} + 96 \, b^{2} c^{4} + 7 \, c^{6}\right )} \sqrt {b^{2} + c^{2}}\right )}}{15 \, {\left (c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + b + \sqrt {b^{2} + c^{2}}\right )}^{5} c^{5} e} \]
[In]
[Out]
Time = 33.51 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.10 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {32\,b^4+32\,b^2\,c^2+4\,c^4}{c^7}+\frac {\left (32\,b^3+16\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^7}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (\frac {8\,b^3+6\,b\,c^2}{c^6}+\frac {\left (8\,b^2+2\,c^2\right )\,\sqrt {b^2+c^2}}{c^6}\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {64\,b^6+\frac {304\,b^4\,c^2}{3}+\frac {124\,b^2\,c^4}{3}+\frac {8\,c^6}{3}}{c^9}+\frac {\sqrt {b^2+c^2}\,\left (64\,b^5+\frac {208\,b^3\,c^2}{3}+\frac {44\,b\,c^4}{3}\right )}{c^9}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {64\,b^5+\frac {256\,b^3\,c^2}{3}+24\,b\,c^4}{c^8}+\frac {\sqrt {b^2+c^2}\,\left (64\,b^4+\frac {160\,b^2\,c^2}{3}+\frac {16\,c^4}{3}\right )}{c^8}\right )+\frac {\frac {128\,b^7}{5}+\frac {704\,b^5\,c^2}{15}+\frac {80\,b^3\,c^4}{3}+\frac {14\,b\,c^6}{3}}{c^{10}}+\frac {\sqrt {b^2+c^2}\,\left (\frac {128\,b^6}{5}+\frac {512\,b^4\,c^2}{15}+\frac {64\,b^2\,c^4}{5}+\frac {14\,c^6}{15}\right )}{c^{10}}}{e\,\left (\frac {16\,b^5+20\,b^3\,c^2+5\,b\,c^4}{c^5}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {40\,b^3+30\,b\,c^2}{c^3}+\frac {\left (40\,b^2+10\,c^2\right )\,\sqrt {b^2+c^2}}{c^3}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {20\,b^2+10\,c^2}{c^2}+\frac {20\,b\,\sqrt {b^2+c^2}}{c^2}\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {40\,b^4+40\,b^2\,c^2+5\,c^4}{c^4}+\frac {\left (40\,b^3+20\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^4}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (\frac {5\,\sqrt {b^2+c^2}}{c}+\frac {5\,b}{c}\right )+\frac {\sqrt {b^2+c^2}\,\left (16\,b^4+12\,b^2\,c^2+c^4\right )}{c^5}\right )} \]
[In]
[Out]