Optimal. Leaf size=509 \[ -\frac {\sqrt {d} \sqrt {g} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f} \]
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Rubi [A]
time = 0.55, antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2988, 2654,
303, 1176, 631, 210, 1179, 642, 2985, 2984, 504, 1232} \begin {gather*} \frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right )}{b f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right )}{b f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}}-\frac {\sqrt {d} \sqrt {g} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)+\sqrt {g}\right )}{2 \sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)+\sqrt {g}\right )}{2 \sqrt {2} b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 504
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1232
Rule 2654
Rule 2984
Rule 2985
Rule 2988
Rubi steps
\begin {align*} \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx &=\frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}} \, dx}{b}-\frac {(a d) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx}{b}\\ &=\frac {\left (2 d^2 g\right ) \text {Subst}\left (\int \frac {x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{b f}-\frac {\left (a d \sqrt {\cos (e+f x)}\right ) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \cos (e+f x))} \, dx}{b \sqrt {d \cos (e+f x)}}\\ &=-\frac {(d g) \text {Subst}\left (\int \frac {g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{b f}+\frac {(d g) \text {Subst}\left (\int \frac {g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{b f}-\frac {\left (4 \sqrt {2} a d g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{b f \sqrt {d \cos (e+f x)}}\\ &=\frac {\left (\sqrt {d} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}+2 x}{-\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{2 \sqrt {2} b f}+\frac {\left (\sqrt {d} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}-2 x}{-\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{2 \sqrt {2} b f}+\frac {g \text {Subst}\left (\int \frac {1}{\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{2 b f}+\frac {g \text {Subst}\left (\int \frac {1}{\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{2 b f}-\frac {\left (2 \sqrt {2} a d g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{b \sqrt {-a+b} f \sqrt {d \cos (e+f x)}}+\frac {\left (2 \sqrt {2} a d g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{b \sqrt {-a+b} f \sqrt {d \cos (e+f x)}}\\ &=\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}+\frac {\left (\sqrt {d} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}-\frac {\left (\sqrt {d} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}\\ &=-\frac {\sqrt {d} \sqrt {g} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 13.34, size = 264, normalized size = 0.52 \begin {gather*} \frac {2 \sqrt {2} g \sqrt {d \cos (e+f x)} \left (i \sqrt {-a-b} \sqrt {a-b} \Pi \left (-i;\left .\text {ArcSin}\left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right )\right |-1\right )-i \sqrt {-a-b} \sqrt {a-b} \Pi \left (i;\left .\text {ArcSin}\left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right )\right |-1\right )+a \left (-\Pi \left (-\frac {\sqrt {a-b}}{\sqrt {-a-b}};\left .\text {ArcSin}\left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right )\right |-1\right )+\Pi \left (\frac {\sqrt {a-b}}{\sqrt {-a-b}};\left .\text {ArcSin}\left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right )\right |-1\right )\right )\right ) \sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}}{\sqrt {-a-b} \sqrt {a-b} b f \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {g \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 879 vs. \(2 (393 ) = 786\).
time = 1.99, size = 880, normalized size = 1.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(880\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d \cos {\left (e + f x \right )}} \sqrt {g \sin {\left (e + f x \right )}}}{a + b \cos {\left (e + f x \right )}}\, dx \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 {\sqrt {d\,\cos \left (e+f\,x\right )}\,\sqrt {g\,\sin \left (e+f\,x\right )}}{a+b\,\cos \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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