Optimal. Leaf size=341 \[ -\frac {a \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \sqrt [4]{-a^2+b^2} f}+\frac {a \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \sqrt [4]{-a^2+b^2} f}+\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {\cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}} \]
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Rubi [A]
time = 0.46, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2946, 2721,
2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} -\frac {a \sqrt {g} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{3/2} f \sqrt [4]{b^2-a^2}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 f \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 f \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}+\frac {a \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{3/2} f \sqrt [4]{b^2-a^2}}+\frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{b f \sqrt {\cos (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2780
Rule 2884
Rule 2886
Rule 2946
Rubi steps
\begin {align*} \int \frac {\sqrt {g \cos (e+f x)} \sin (e+f x)}{a+b \sin (e+f x)} \, dx &=\frac {\int \sqrt {g \cos (e+f x)} \, dx}{b}-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{b}\\ &=\frac {\left (a^2 g\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^2}-\frac {\left (a^2 g\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^2}-\frac {(a g) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{f}+\frac {\sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)} \, dx}{b \sqrt {\cos (e+f x)}}\\ &=\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {\cos (e+f x)}}-\frac {(2 a g) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{f}+\frac {\left (a^2 g \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^2 \sqrt {g \cos (e+f x)}}-\frac {\left (a^2 g \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^2 \sqrt {g \cos (e+f x)}}\\ &=\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {\cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {(a g) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b f}-\frac {(a g) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b f}\\ &=-\frac {a \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \sqrt [4]{-a^2+b^2} f}+\frac {a \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \sqrt [4]{-a^2+b^2} f}+\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {\cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^2 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 16.28, size = 351, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {g \cos (e+f x)} \left (8 b^{5/2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )\right ) \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{12 b^{3/2} \left (-a^2+b^2\right ) f \sqrt {\cos (e+f x)} (a+b \sin (e+f x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 42.01, size = 693, normalized size = 2.03
method | result | size |
default | \(\frac {-\frac {g a \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{8}-4 b^{2} g \,\textit {\_Z}^{6}+\left (16 a^{2} g^{2}-10 b^{2} g^{2}\right ) \textit {\_Z}^{4}-4 b^{2} g^{3} \textit {\_Z}^{2}+b^{2} g^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-\textit {\_R}^{4} g -\textit {\_R}^{2} g^{2}+g^{3}\right ) \ln \left (\sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}-\sqrt {g}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sqrt {2}-\textit {\_R} \right )}{\textit {\_R}^{7} b^{2}-3 \textit {\_R}^{5} b^{2} g +8 \textit {\_R}^{3} a^{2} g^{2}-5 \textit {\_R}^{3} b^{2} g^{2}-\textit {\_R} \,b^{2} g^{3}}\right )}{2}-\frac {8 \sqrt {g \left (2 \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\, g b \left (\frac {\left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (f x +e \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{2 b^{2} \sqrt {-g \left (2 \left (\sin ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (4 b^{2} \textit {\_Z}^{4}-4 b^{2} \textit {\_Z}^{2}+a^{2}\right )}{\sum }\frac {\left (-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+\left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-a^{2}\right ) \left (\frac {\sqrt {2}\, \arctanh \left (\frac {g \left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}-3\right ) \left (4 \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}-3 b^{2} \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-3 a^{2}+2 b^{2}\right ) \sqrt {2}}{2 \left (4 a^{2}-3 b^{2}\right ) \sqrt {\frac {g \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}\, \sqrt {-g \left (2 \left (\sin ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}}\right )}{\sqrt {\frac {g \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}}+\frac {8 b^{2} \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (f x +e \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right ), -\frac {4 b^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )}{a^{2}}, \sqrt {2}\right )}{a^{2} \sqrt {-\left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g \left (2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )}}{32 b^{4}}\right )}{\sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \sqrt {g \left (2 \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-1\right )}}}{f}\) | \(693\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (e+f\,x\right )\,\sqrt {g\,\cos \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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