Optimal. Leaf size=352 \[ \frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\sqrt {b} \left (-a^2+b^2\right )^{3/4} f \sqrt {g}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\sqrt {b} \left (-a^2+b^2\right )^{3/4} f \sqrt {g}}+\frac {2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b \left (a^2-b^2+b \sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) f \sqrt {g \cos (e+f x)}} \]
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Rubi [A]
time = 0.47, antiderivative size = 352, 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,
2720, 2781, 2886, 2884, 335, 218, 214, 211} \begin {gather*} \frac {a \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt {b} f \sqrt {g} \left (b^2-a^2\right )^{3/4}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt {b} f \sqrt {g} \left (b^2-a^2\right )^{3/4}}-\frac {a^2 \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 f \left (b \sqrt {b^2-a^2}+a^2-b^2\right ) \sqrt {g \cos (e+f x)}}-\frac {a^2 \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 f \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {g \cos (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2781
Rule 2884
Rule 2886
Rule 2946
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx &=\frac {\int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{b}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b}\\ &=\frac {a^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b \sqrt {-a^2+b^2}}+\frac {a^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b \sqrt {-a^2+b^2}}-\frac {(a g) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) g^2+b^2 x^2\right )} \, dx,x,g \cos (e+f x)\right )}{f}+\frac {\sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{b \sqrt {g \cos (e+f x)}}\\ &=\frac {2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {(2 a g) \text {Subst}\left (\int \frac {1}{\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 \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 \sqrt {-a^2+b^2} \sqrt {g \cos (e+f x)}}+\frac {\left (a^2 \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 \sqrt {-a^2+b^2} \sqrt {g \cos (e+f x)}}\\ &=\frac {2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a^2 \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 \left (a^2-b^2+b \sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a^2 \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 \sqrt {-a^2+b^2} \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{\sqrt {-a^2+b^2} f}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{\sqrt {-a^2+b^2} f}\\ &=\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\sqrt {b} \left (-a^2+b^2\right )^{3/4} f \sqrt {g}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\sqrt {b} \left (-a^2+b^2\right )^{3/4} f \sqrt {g}}+\frac {2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a^2 \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 \left (a^2-b^2+b \sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a^2 \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 \sqrt {-a^2+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 = 9.81, size = 546, normalized size = 1.55 \begin {gather*} -\frac {2 \sqrt {\cos (e+f x)} \left (a+b \sqrt {\sin ^2(e+f x)}\right ) \left (\frac {a \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 )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \sqrt {\cos (e+f x)} \sqrt {\sin ^2(e+f x)}}{\left (a^2-b^2+b^2 \cos ^2(e+f x)\right ) \left (-5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+2 \left (2 b^2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)\right )}\right )}{f \sqrt {g \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 = 38.14, size = 687, normalized size = 1.95
method | result | size |
default | \(\frac {-2 a g \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}^{4}+\textit {\_R}^{2} g \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 )-\frac {\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 )}\, \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \left (\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 {\underline {\hspace {1.25 ex}}\alpha \left (8 \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 ) \sqrt {\frac {g \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-8 b^{2} \underline {\hspace {1.25 ex}}\alpha \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 ) \sqrt {\frac {g \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}+\sqrt {2}\, a^{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 {-\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 )}{\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \sqrt {\frac {g \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{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 )}{2 b \,a^{2} \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}\) | \(687\) |
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 )}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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