Optimal. Leaf size=298 \[ -\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))}-\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a^3 f (c-d)^4 (c+d) \sqrt {c^2-d^2}}-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))} \]
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Rubi [A] time = 0.73, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2766, 2978, 2754, 12, 2660, 618, 204} \[ -\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a^3 f (c-d)^4 (c+d) \sqrt {c^2-d^2}}-\frac {d \left (-12 c^2 d+2 c^3+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 2660
Rule 2754
Rule 2766
Rule 2978
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {\int \frac {-2 a (c-3 d)-3 a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx}{5 a^2 (c-d)}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}+\frac {\int \frac {a^2 \left (2 c^2-8 c d+27 d^2\right )+2 a^2 (2 c-9 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\int \frac {-2 a^3 (c-36 d) d^2-a^3 d \left (2 c^2-12 c d+45 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}+\frac {\int -\frac {15 a^3 d^3 (4 c+3 d)}{c+d \sin (e+f x)} \, dx}{15 a^6 (c-d)^4 (c+d)}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\left (d^3 (4 c+3 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{a^3 (c-d)^4 (c+d)}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\left (2 d^3 (4 c+3 d)\right ) \operatorname {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^3 (c-d)^4 (c+d) f}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}+\frac {\left (4 d^3 (4 c+3 d)\right ) \operatorname {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^3 (c-d)^4 (c+d) f}\\ &=-\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^3 (c-d)^4 (c+d) \sqrt {c^2-d^2} f}-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 2.55, size = 361, normalized size = 1.21 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 \left (2 c^2-14 c d+57 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4-\frac {30 d^3 (4 c+3 d) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{(c+d) \sqrt {c^2-d^2}}-\frac {15 d^4 \cos (e+f x) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d) (c+d \sin (e+f x))}+6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-2 (c-6 d) (c-d) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3+4 (c-6 d) (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2-3 (c-d)^2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{15 a^3 f (c-d)^4 (\sin (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 3235, normalized size = 10.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 517, normalized size = 1.73 \[ -\frac {2 \, {\left (\frac {15 \, {\left (4 \, c d^{3} + 3 \, d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (c) + \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^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {15 \, {\left (d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c d^{4}\right )}}{{\left (a^{3} c^{6} - 3 \, a^{3} c^{5} d + 2 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} - 3 \, a^{3} c^{2} d^{4} + a^{3} c 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 )}} + \frac {15 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 90 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 150 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 300 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 190 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 420 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 110 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 270 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{2} - 34 \, c d + 72 \, d^{2}}{{\left (a^{3} c^{4} - 4 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} - 4 \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}\right )}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 511, normalized size = 1.71 \[ -\frac {2 d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a^{3} f \left (c -d \right )^{4} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c +d \right ) c}-\frac {2 d^{4}}{a^{3} f \left (c -d \right )^{4} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c +d \right )}-\frac {8 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) c}{a^{3} f \left (c -d \right )^{4} \left (c +d \right ) \sqrt {c^{2}-d^{2}}}-\frac {6 d^{4} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{a^{3} f \left (c -d \right )^{4} \left (c +d \right ) \sqrt {c^{2}-d^{2}}}-\frac {16 c}{3 a^{3} f \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {8 d}{a^{3} f \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8 d}{a^{3} f \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {4 c}{a^{3} f \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 c^{2}}{a^{3} f \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {8 c d}{a^{3} f \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {12 d^{2}}{a^{3} f \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{5 a^{3} f \left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{a^{3} f \left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.30, size = 987, normalized size = 3.31 \[ -\frac {\frac {2\,\left (7\,c^4-27\,c^3\,d+38\,c^2\,d^2+72\,c\,d^3+15\,d^4\right )}{15\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (5\,c^4-18\,c^3\,d+19\,c^2\,d^2+84\,c\,d^3+15\,d^4\right )}{3\,c\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^5-76\,c^4\,d+106\,c^3\,d^2+346\,c^2\,d^3+219\,c\,d^4+15\,d^5\right )}{15\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (c^5-3\,c^4\,d+2\,c^3\,d^2+6\,c^2\,d^3+d^5\right )}{c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,c^5-6\,c^4\,d+4\,c^3\,d^2+24\,c^2\,d^3+13\,c\,d^4+5\,d^5\right )}{c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (11\,c^5-27\,c^4\,d+4\,c^3\,d^2+162\,c^2\,d^3+135\,c\,d^4+30\,d^5\right )}{3\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (47\,c^5-137\,c^4\,d+88\,c^3\,d^2+812\,c^2\,d^3+690\,c\,d^4+75\,d^5\right )}{15\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}}{f\,\left (a^3\,c+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,a^3\,c+2\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (5\,a^3\,c+2\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,a^3\,c+10\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (11\,a^3\,c+10\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (15\,a^3\,c+20\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (15\,a^3\,c+20\,a^3\,d\right )+a^3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}-\frac {2\,d^3\,\mathrm {atan}\left (\frac {\frac {d^3\,\left (4\,c+3\,d\right )\,\left (2\,a^3\,c^5\,d-6\,a^3\,c^4\,d^2+4\,a^3\,c^3\,d^3+4\,a^3\,c^2\,d^4-6\,a^3\,c\,d^5+2\,a^3\,d^6\right )}{a^3\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}}+\frac {2\,c\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c+3\,d\right )\,\left (a^3\,c^5-3\,a^3\,c^4\,d+2\,a^3\,c^3\,d^2+2\,a^3\,c^2\,d^3-3\,a^3\,c\,d^4+a^3\,d^5\right )}{a^3\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}}}{6\,d^4+8\,c\,d^3}\right )\,\left (4\,c+3\,d\right )}{a^3\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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