Optimal. Leaf size=192 \[ \frac {2 \sqrt {a+b} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{\sqrt {c+d} (b c-a d) f} \]
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Rubi [A]
time = 0.07, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2897}
\begin {gather*} \frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2897
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx &=\frac {2 \sqrt {a+b} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{\sqrt {c+d} (b c-a d) f}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 191, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {a+b} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {\frac {(-b c+a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{\sqrt {c+d} (b c-a d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1235\) vs.
\(2(177)=354\).
time = 12.99, size = 1236, normalized size = 6.44
method | result | size |
default | \(-\frac {4 \EllipticF \left (\sqrt {-\frac {\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )-c \sin \left (f x +e \right )-d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}-d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}, \sqrt {\frac {\left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right )}{\left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\right ) \sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}-a \sin \left (f x +e \right )-\cos \left (f x +e \right ) b +\sqrt {-a^{2}+b^{2}}-b \right ) \sqrt {-c^{2}+d^{2}}\, c}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right )}}\, \sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+\cos \left (f x +e \right ) b +\sqrt {-a^{2}+b^{2}}+b \right ) \sqrt {-c^{2}+d^{2}}\, c}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\, \sqrt {-\frac {\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )-c \sin \left (f x +e \right )-d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}-d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\, \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}\, d -\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, c^{2}+\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, d^{2}-\cos \left (f x +e \right ) \sqrt {-c^{2}+d^{2}}\, a c +\cos \left (f x +e \right ) \sqrt {-c^{2}+d^{2}}\, b d -\cos \left (f x +e \right ) b \,c^{2}+\cos \left (f x +e \right ) b \,d^{2}+c \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}\, \sin \left (f x +e \right )+c d \sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b c \sqrt {-c^{2}+d^{2}}\, \sin \left (f x +e \right )-a \,c^{2} \sin \left (f x +e \right )+b c d \sin \left (f x +e \right )+d \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}+d^{2} \sqrt {-a^{2}+b^{2}}+b d \sqrt {-c^{2}+d^{2}}-a c d +d^{2} b \right )}{f \sin \left (f x +e \right )^{4} \left (\left (\cos ^{2}\left (f x +e \right )\right ) b d -\sin \left (f x +e \right ) a d -\sin \left (f x +e \right ) b c -a c -b d \right ) \sqrt {-c^{2}+d^{2}}\, \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}\) | \(1236\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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