Integrand size = 25, antiderivative size = 204 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {d \left (2 c^2+2 c d+d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) \left (c^2-d^2\right )^{5/2} f}-\frac {d (2 c+3 d) \cos (e+f x)}{6 (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (3+3 \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{6 (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))} \]
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Time = 0.38 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2847, 2833, 12, 2739, 632, 210} \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {3 d \left (2 c^2+2 c d+d^2\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a f (c-d) \left (c^2-d^2\right )^{5/2}}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac {d (2 c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2847
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {d \int \frac {-3 a+2 a \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{a^2 (c-d)} \\ & = -\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \int \frac {2 a (3 c+2 d)-a (2 c+3 d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 a^2 (c-d)^2 (c+d)} \\ & = -\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}+\frac {d \int -\frac {3 a \left (2 c^2+2 c d+d^2\right )}{c+d \sin (e+f x)} \, dx}{2 a^2 (c-d)^3 (c+d)^2} \\ & = -\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (3 d \left (2 c^2+2 c d+d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 a (c-d)^3 (c+d)^2} \\ & = -\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (3 d \left (2 c^2+2 c d+d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a (c-d)^3 (c+d)^2 f} \\ & = -\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (6 d \left (2 c^2+2 c d+d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{a (c-d)^3 (c+d)^2 f} \\ & = -\frac {3 d \left (2 c^2+2 c d+d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a (c-d)^3 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d (2 c+3 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+4 d) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {\cos (e+f x) \left (-\frac {6 d \left (2 c^2+2 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{(-c-d)^{3/2} (c-d)^{5/2} \sqrt {\cos ^2(e+f x)}}+\frac {2 c^2+9 c d+4 d^2}{(c-d)^2 (c+d) (1+\sin (e+f x))}-\frac {d}{(1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d (4 c+d)}{(c-d) (c+d) (1+\sin (e+f x)) (c+d \sin (e+f x))}\right )}{6 (-c+d) (c+d) f} \]
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Time = 2.27 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {-\frac {2}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2 d \left (\frac {\frac {d^{2} \left (7 c^{2}+2 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 c^{4}+2 c^{3} d +11 c^{2} d^{2}+4 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (17 c^{2}+6 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (6 c^{2}+2 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {3 \left (2 c^{2}+2 c d +d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{3}}}{f a}\) | \(327\) |
default | \(\frac {-\frac {2}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2 d \left (\frac {\frac {d^{2} \left (7 c^{2}+2 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 c^{4}+2 c^{3} d +11 c^{2} d^{2}+4 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (17 c^{2}+6 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (6 c^{2}+2 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {3 \left (2 c^{2}+2 c d +d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{3}}}{f a}\) | \(327\) |
risch | \(-\frac {8 i c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}-3 i d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-18 i c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}-15 i c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-6 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-6 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-3 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+i d^{4} {\mathrm e}^{i \left (f x +e \right )}+19 i c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+32 i c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}-24 i c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+8 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+26 d \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+32 d^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+19 d^{3} c \,{\mathrm e}^{2 i \left (f x +e \right )}+5 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-2 c^{2} d^{2}-9 d^{3} c -4 d^{4}}{\left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{2} \left (c +d \right )^{2} \left (c -d \right )^{3} f a}-\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}-\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}+\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}+\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right )^{3} f a}\) | \(847\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1140 vs. \(2 (204) = 408\).
Time = 0.35 (sec) , antiderivative size = 2365, normalized size of antiderivative = 11.59 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (204) = 408\).
Time = 0.36 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {\frac {3 \, {\left (2 \, c^{2} d + 2 \, c d^{2} + d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (a c^{5} - a c^{4} d - 2 \, a c^{3} d^{2} + 2 \, a c^{2} d^{3} + a c d^{4} - a d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {7 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 17 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, c^{4} d^{2} + 2 \, c^{3} d^{3} - c^{2} d^{4}}{{\left (a c^{7} - a c^{6} d - 2 \, a c^{5} d^{2} + 2 \, a c^{4} d^{3} + a c^{3} d^{4} - a c^{2} d^{5}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}} + \frac {2}{{\left (a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}}{f} \]
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Time = 10.32 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.69 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {3\,d\,\mathrm {atan}\left (\frac {\frac {3\,d\,\left (2\,c^2+2\,c\,d+d^2\right )\,\left (-2\,a\,c^5\,d+2\,a\,c^4\,d^2+4\,a\,c^3\,d^3-4\,a\,c^2\,d^4-2\,a\,c\,d^5+2\,a\,d^6\right )}{2\,a\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}}-\frac {3\,c\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^2+2\,c\,d+d^2\right )\,\left (a\,c^5-a\,c^4\,d-2\,a\,c^3\,d^2+2\,a\,c^2\,d^3+a\,c\,d^4-a\,d^5\right )}{a\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}}}{6\,c^2\,d+6\,c\,d^2+3\,d^3}\right )\,\left (2\,c^2+2\,c\,d+d^2\right )}{a\,f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}}-\frac {\frac {2\,c^4+4\,c^3\,d+8\,c^2\,d^2+2\,c\,d^3-d^4}{\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (8\,c^5\,d+22\,c^4\,d^2+17\,c^3\,d^3+13\,c^2\,d^4+2\,c\,d^5-2\,d^6\right )}{c^2\,\left (c^2-2\,c\,d+d^2\right )\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (4\,c^5+4\,c^4\,d+14\,c^3\,d^2+21\,c^2\,d^3+4\,c\,d^4-2\,d^5\right )}{c^2\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^5+4\,c^4\,d+2\,c^3\,d^2+7\,c^2\,d^3+2\,c\,d^4-2\,d^5\right )}{c\,\left (c^2-2\,c\,d+d^2\right )\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c^4\,d+22\,c^3\,d^2+27\,c^2\,d^3+5\,c\,d^4-2\,d^5\right )}{c\,\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a\,c^2+4\,a\,c\,d+4\,a\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a\,c^2+4\,a\,c\,d+4\,a\,d^2\right )+a\,c^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^2+4\,a\,d\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (a\,c^2+4\,a\,d\,c\right )+a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )} \]
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