Optimal. Leaf size=129 \[ \frac {b^2 x}{d^2}-\frac {2 (b c-a d) \left (a c d+b \left (c^2-2 d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^2 \left (c^2-d^2\right )^{3/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.19, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2869, 2814,
2739, 632, 210} \begin {gather*} -\frac {2 (b c-a d) \left (a c d+b \left (c^2-2 d^2\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^2 f \left (c^2-d^2\right )^{3/2}}+\frac {(b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) (c+d \sin (e+f x))}+\frac {b^2 x}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2869
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx &=\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {\int \frac {d \left (\left (a^2+b^2\right ) c-2 a b d\right )+b^2 \left (c^2-d^2\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{d \left (c^2-d^2\right )}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))}-\frac {\left (-d^2 \left (\left (a^2+b^2\right ) c-2 a b d\right )+b^2 c \left (c^2-d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^2 \left (c^2-d^2\right )}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))}-\frac {\left (2 \left (-d^2 \left (\left (a^2+b^2\right ) c-2 a b d\right )+b^2 c \left (c^2-d^2\right )\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 )}{d^2 \left (c^2-d^2\right ) f}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {\left (4 \left (-d^2 \left (\left (a^2+b^2\right ) c-2 a b d\right )+b^2 c \left (c^2-d^2\right )\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 )}{d^2 \left (c^2-d^2\right ) f}\\ &=\frac {b^2 x}{d^2}-\frac {2 (b c-a d) \left (b c^2+a c d-2 b d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^2 \left (c^2-d^2\right )^{3/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 134, normalized size = 1.04 \begin {gather*} \frac {b^2 (e+f x)-\frac {2 \left (-a^2 c d^2+2 a b d^3+b^2 \left (c^3-2 c d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {d (b c-a d)^2 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{d^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 218, normalized size = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {2 \left (\frac {d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{c^{2}-d^{2}}\right )}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (a^{2} c \,d^{2}-2 a b \,d^{3}-b^{2} c^{3}+2 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{2}}}{f}\) | \(218\) |
default | \(\frac {\frac {2 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {2 \left (\frac {d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{c^{2}-d^{2}}\right )}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (a^{2} c \,d^{2}-2 a b \,d^{3}-b^{2} c^{3}+2 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{2}}}{f}\) | \(218\) |
risch | \(\frac {b^{2} x}{d^{2}}-\frac {2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (i d +c \,{\mathrm e}^{i \left (f x +e \right )}\right )}{d^{2} \left (c^{2}-d^{2}\right ) f \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c -c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c -c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a b}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c -c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c^{3}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,d^{2}}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c -c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c +c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c +c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a b}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c +c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c^{3}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,d^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-c^{2}+d^{2}}\, c +c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}\) | \(761\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs.
\(2 (127) = 254\).
time = 0.40, size = 694, normalized size = 5.38 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right )} f x \sin \left (f x + e\right ) + 2 \, {\left (b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right )} f x - {\left (b^{2} c^{4} + 2 \, a b c d^{3} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d^{2} + {\left (b^{2} c^{3} d + 2 \, a b d^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (-\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} - 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right )} f\right )}}, \frac {{\left (b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right )} f x \sin \left (f x + e\right ) + {\left (b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right )} f x + {\left (b^{2} c^{4} + 2 \, a b c d^{3} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d^{2} + {\left (b^{2} c^{3} d + 2 \, a b d^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} \cos \left (f x + e\right )}{{\left (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right )} f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 249, normalized size = 1.93 \begin {gather*} \frac {\frac {{\left (f x + e\right )} b^{2}}{d^{2}} - \frac {2 \, {\left (b^{2} c^{3} - a^{2} c d^{2} - 2 \, b^{2} c d^{2} + 2 \, a b 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 (c^{2} d^{2} - d^{4}\right )} \sqrt {c^{2} - d^{2}}} + \frac {2 \, {\left (b^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}}{{\left (c^{3} d - c d^{3}\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 )}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.47, size = 2500, normalized size = 19.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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