Optimal. Leaf size=162 \[ \frac {\left (2 a^2 c-3 a b d+b^2 c\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{5/2}}+\frac {\left (a^2 (-d)+3 a b c-2 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {(b c-a d) \cos (e+f x)}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2} \]
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Rubi [A] time = 0.17, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 12, 2660, 618, 204} \[ \frac {\left (2 a^2 c-3 a b d+b^2 c\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{5/2}}+\frac {\left (a^2 (-d)+3 a b c-2 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {(b c-a d) \cos (e+f x)}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 2660
Rule 2754
Rubi steps
\begin {align*} \int \frac {c+d \sin (e+f x)}{(a+b \sin (e+f x))^3} \, dx &=\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-2 (a c-b d)+(b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 a b c-a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\int \frac {2 a^2 c+b^2 c-3 a b d}{a+b \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 a b c-a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\left (2 a^2 c+b^2 c-3 a b d\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 a b c-a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\left (2 a^2 c+b^2 c-3 a b d\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 f}\\ &=\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 a b c-a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}-\frac {\left (2 \left (2 a^2 c+b^2 c-3 a b d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 f}\\ &=\frac {\left (2 a^2 c+b^2 c-3 a b d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} f}+\frac {(b c-a d) \cos (e+f x)}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 a b c-a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 157, normalized size = 0.97 \[ \frac {\frac {2 \left (2 a^2 c-3 a b d+b^2 c\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2 d-3 a b c+2 b^2 d\right ) \cos (e+f x)}{(a-b)^2 (a+b)^2 (a+b \sin (e+f x))}+\frac {(b c-a d) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))^2}}{2 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 799, normalized size = 4.93 \[ \left [-\frac {2 \, {\left (3 \, {\left (a^{3} b^{2} - a b^{4}\right )} c - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (3 \, a b^{3} d - {\left (2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right )} c - 3 \, {\left (a^{3} b + a b^{3}\right )} d - 2 \, {\left (3 \, a^{2} b^{2} d - {\left (2 \, a^{3} b + a b^{3}\right )} c\right )} \sin \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left ({\left (4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right )} c - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d\right )} \cos \left (f x + e\right )}{4 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} f \sin \left (f x + e\right ) - {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} f\right )}}, -\frac {{\left (3 \, {\left (a^{3} b^{2} - a b^{4}\right )} c - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left ({\left (3 \, a b^{3} d - {\left (2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right )} c - 3 \, {\left (a^{3} b + a b^{3}\right )} d - 2 \, {\left (3 \, a^{2} b^{2} d - {\left (2 \, a^{3} b + a b^{3}\right )} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + {\left ({\left (4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right )} c - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d\right )} \cos \left (f x + e\right )}{2 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} f \sin \left (f x + e\right ) - {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 429, normalized size = 2.65 \[ \frac {\frac {{\left (2 \, a^{2} c + b^{2} c - 3 \, a b d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {5 \, a^{3} b^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a b^{4} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} b d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a^{4} b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{2} b^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, b^{5} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5 \, a^{3} b^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a b^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, a^{3} b^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b^{4} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{4} b d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a^{2} b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, a^{4} b c - a^{2} b^{3} c - 2 \, a^{5} d - a^{3} b^{2} d}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 1291, normalized size = 7.97 \[ \text {result too large to display} \]
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 = 9.79, size = 477, normalized size = 2.94 \[ \frac {\mathrm {atan}\left (\frac {\left (\frac {\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )\,\left (2\,c\,a^2-3\,d\,a\,b+c\,b^2\right )}{2\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c\,a^2-3\,d\,a\,b+c\,b^2\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{2\,c\,a^2-3\,d\,a\,b+c\,b^2}\right )\,\left (2\,c\,a^2-3\,d\,a\,b+c\,b^2\right )}{f\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}-\frac {\frac {2\,d\,a^3-4\,c\,a^2\,b+d\,a\,b^2+c\,b^3}{a^4-2\,a^2\,b^2+b^4}+\frac {b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,d\,a^3-5\,c\,a^2\,b+2\,c\,b^3\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,d\,a^3-11\,c\,a^2\,b+4\,d\,a\,b^2+2\,c\,b^3\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^2+2\,b^2\right )\,\left (2\,d\,a^3-4\,c\,a^2\,b+d\,a\,b^2+c\,b^3\right )}{a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^2+4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+a^2+4\,a\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,a\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )} \]
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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