Optimal. Leaf size=65 \[ \frac {d x}{b}+\frac {2 (b c-a d) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2} f} \]
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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2814, 2739,
632, 210} \begin {gather*} \frac {2 (b c-a d) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b f \sqrt {a^2-b^2}}+\frac {d x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rubi steps
\begin {align*} \int \frac {c+d \sin (e+f x)}{a+b \sin (e+f x)} \, dx &=\frac {d x}{b}-\frac {(-b c+a d) \int \frac {1}{a+b \sin (e+f x)} \, dx}{b}\\ &=\frac {d x}{b}+\frac {(2 (b c-a d)) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b f}\\ &=\frac {d x}{b}-\frac {(4 (b c-a d)) \text {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 )}{b f}\\ &=\frac {d x}{b}+\frac {2 (b c-a d) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2} f}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 67, normalized size = 1.03 \begin {gather*} \frac {d (e+f x)+\frac {2 (b c-a d) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}}{b f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 76, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {\frac {2 d \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b}+\frac {2 \left (-a d +b c \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}}}{f}\) | \(76\) |
default | \(\frac {\frac {2 d \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b}+\frac {2 \left (-a d +b c \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}}}{f}\) | \(76\) |
risch | \(\frac {d x}{b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a d}{\sqrt {-a^{2}+b^{2}}\, f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a d}{\sqrt {-a^{2}+b^{2}}\, f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c}{\sqrt {-a^{2}+b^{2}}\, f}\) | \(282\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 262, normalized size = 4.03 \begin {gather*} \left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d f x + \sqrt {-a^{2} + b^{2}} {\left (b c - a d\right )} \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 (a^{2} b - b^{3}\right )} f}, \frac {{\left (a^{2} - b^{2}\right )} d f x - \sqrt {a^{2} - b^{2}} {\left (b c - a d\right )} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right )}{{\left (a^{2} b - b^{3}\right )} f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 537 vs.
\(2 (53) = 106\).
time = 41.16, size = 537, normalized size = 8.26 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x \left (c + d \sin {\left (e \right )}\right )}{\sin {\left (e \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\\frac {c x - \frac {d \cos {\left (e + f x \right )}}{f}}{a} & \text {for}\: b = 0 \\\frac {x \left (c + d \sin {\left (e \right )}\right )}{a + b \sin {\left (e \right )}} & \text {for}\: f = 0 \\\frac {b^{2} d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - f \left (b^{2}\right )^{\frac {3}{2}}} + \frac {2 b^{2} d}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - f \left (b^{2}\right )^{\frac {3}{2}}} + \frac {2 b c \sqrt {b^{2}}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - f \left (b^{2}\right )^{\frac {3}{2}}} - \frac {b d f x \sqrt {b^{2}}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - f \left (b^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = - \sqrt {b^{2}} \\\frac {b^{2} d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + f \left (b^{2}\right )^{\frac {3}{2}}} + \frac {2 b^{2} d}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + f \left (b^{2}\right )^{\frac {3}{2}}} - \frac {2 b c \sqrt {b^{2}}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + f \left (b^{2}\right )^{\frac {3}{2}}} + \frac {b d f x \sqrt {b^{2}}}{b^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + f \left (b^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = \sqrt {b^{2}} \\\frac {\frac {c \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} \right )}}{f} + d x}{b} & \text {for}\: a = 0 \\- \frac {a d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b f \sqrt {- a^{2} + b^{2}}} + \frac {a d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b f \sqrt {- a^{2} + b^{2}}} + \frac {c \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{f \sqrt {- a^{2} + b^{2}}} - \frac {c \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{f \sqrt {- a^{2} + b^{2}}} + \frac {d x}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 86, normalized size = 1.32 \begin {gather*} \frac {\frac {{\left (f x + e\right )} d}{b} + \frac {2 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \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 (b c - a d\right )}}{\sqrt {a^{2} - b^{2}} b}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.01, size = 343, normalized size = 5.28 \begin {gather*} \frac {2\,d\,\mathrm {atan}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{b\,f}-\frac {a\,\left (d\,\ln \left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {b^2-a^2}}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}-d\,\ln \left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {b^2-a^2}}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,\sqrt {b^2-a^2}\right )-b\,c\,\ln \left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {b^2-a^2}}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}+b\,c\,\ln \left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {b^2-a^2}}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,\sqrt {b^2-a^2}}{b\,f\,\left (a^2-b^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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