Optimal. Leaf size=299 \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{3/4} d \sqrt {e}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{3/4} d \sqrt {e}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.37, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2781, 2886,
2884, 335, 218, 214, 211} \begin {gather*} -\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d \sqrt {e} \left (b^2-a^2\right )^{3/4}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d \sqrt {e} \left (b^2-a^2\right )^{3/4}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \cos (c+d x)}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2781
Rule 2884
Rule 2886
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx &=-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 \sqrt {-a^2+b^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 \sqrt {-a^2+b^2}}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{d}\\ &=\frac {(2 b e) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{d}-\frac {\left (a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}-\frac {\left (a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}\\ &=\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{\sqrt {-a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{\sqrt {-a^2+b^2} d}\\ &=-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{3/4} d \sqrt {e}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{3/4} d \sqrt {e}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d 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 = 26.31, size = 558, normalized size = 1.87 \begin {gather*} -\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x) \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )-\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\left (-a^2+b^2\right )^{3/4}}+\frac {5 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\cos (c+d x)}}{\left (a^2-b^2+b^2 \cos ^2(c+d 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(c+d x),\frac {b^2 \cos ^2(c+d 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(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )\right ) \cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}\right ) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{d \sqrt {e \cos (c+d x)} \sqrt {\sin ^2(c+d x)} (a+b \sin (c+d 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 = 8.36, size = 676, normalized size = 2.26
method | result | size |
default | \(\frac {2 b e \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{8}-4 b^{2} e \,\textit {\_Z}^{6}+\left (16 e^{2} a^{2}-10 b^{2} e^{2}\right ) \textit {\_Z}^{4}-4 b^{2} e^{3} \textit {\_Z}^{2}+b^{2} e^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\textit {\_R}^{2} e \right ) \ln \left (\sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}-\sqrt {e}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}-\textit {\_R} \right )}{\textit {\_R}^{7} b^{2}-3 \textit {\_R}^{5} b^{2} e +8 \textit {\_R}^{3} a^{2} e^{2}-5 \textit {\_R}^{3} b^{2} e^{2}-\textit {\_R} \,b^{2} e^{3}}\right )-\frac {\sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\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 {8 \sqrt {\frac {e \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), -\frac {4 b^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )}{a^{2}}, \sqrt {2}\right ) \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 (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), -\frac {4 b^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )}{a^{2}}, \sqrt {2}\right ) \sqrt {\frac {e \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}+a^{2} \sqrt {2}\, \arctanh \left (\frac {e \left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}-3\right ) \left (4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-3 b^{2} \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{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 {e \left (2 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+a^{2}-2 b^{2}\right )}{b^{2}}}\, \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right ) \sqrt {-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \sqrt {\frac {e \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 {d x}{2}+\frac {c}{2}\right )\right ) e \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}}\right )}{8 a \,b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}}}{d}\) | \(676\) |
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 {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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