Optimal. Leaf size=169 \[ -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2833, 2831,
2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}+\frac {2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2833
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx &=-\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {-\frac {1}{2} a (c-d)+\frac {1}{2} a (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{c^2-d^2}\\ &=-\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {a \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{d}-\frac {a \int \sqrt {c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (a \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{d (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (a \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (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 = 6.29, size = 938, normalized size = 5.55 \begin {gather*} a \left (\frac {(1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)} \left (-\frac {2 \csc (e) \sec (e)}{d (c+d) f}+\frac {2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}\right )}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {\sec (e) (1+\sin (e+f x)) \left (-\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac {\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)}}}{\sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{(c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {2 F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}}{d (c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 3.91, size = 246, normalized size = 1.46
method | result | size |
default | \(\frac {2 \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}+d^{2} \left (\sin ^{2}\left (f x +e \right )\right )-d^{2}\right ) a}{d^{2} \left (c +d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(246\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 543, normalized size = 3.21 \begin {gather*} -\frac {6 \, \sqrt {d \sin \left (f x + e\right ) + c} a d^{2} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - {\left (\sqrt {2} {\left (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (-i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) - i \, \sqrt {2} a c d\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) + i \, \sqrt {2} a c d\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}{3 \, {\left ({\left (c d^{3} + d^{4}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{2} + c d^{3}\right )} f\right )}} \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 = -0.01 \begin {gather*} \int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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