Optimal. Leaf size=203 \[ -\frac {(c-d) (A c+3 B c+7 A d-11 B d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d (3 A c-15 B c-9 A d+13 B d) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 A-7 B) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.39, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3056, 3047,
3102, 2830, 2728, 212} \begin {gather*} -\frac {(c-d) (A c+7 A d+3 B c-11 B d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d^2 (3 A-7 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 a^2 f}+\frac {d (3 A c-9 A d-15 B c+13 B d) \cos (e+f x)}{3 a f \sqrt {a \sin (e+f x)+a}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2830
Rule 3047
Rule 3056
Rule 3102
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{3/2}} \, dx &=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {(c+d \sin (e+f x)) \left (\frac {1}{2} a (A c+3 B c+4 A d-4 B d)-\frac {1}{2} a (3 A-7 B) d \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\frac {1}{2} a c (A c+3 B c+4 A d-4 B d)+\left (-\frac {1}{2} a (3 A-7 B) c d+\frac {1}{2} a d (A c+3 B c+4 A d-4 B d)\right ) \sin (e+f x)-\frac {1}{2} a (3 A-7 B) d^2 \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=\frac {(3 A-7 B) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {-\frac {1}{4} a^2 \left ((3 A-7 B) d^2-3 c (A c+3 B c+4 A d-4 B d)\right )-\frac {1}{2} a^2 d (3 A c-15 B c-9 A d+13 B d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a^3}\\ &=\frac {d (3 A c-15 B c-9 A d+13 B d) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 A-7 B) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {((c-d) (A c+3 B c+7 A d-11 B d)) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a}\\ &=\frac {d (3 A c-15 B c-9 A d+13 B d) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 A-7 B) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {((c-d) (A c+3 B c+7 A d-11 B d)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a f}\\ &=-\frac {(c-d) (A c+3 B c+7 A d-11 B d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d (3 A c-15 B c-9 A d+13 B d) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 A-7 B) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.51, size = 357, normalized size = 1.76 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 (A-B) (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-3 (A-B) (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(3+3 i) (-1)^{3/4} (c-d) (A c+3 B c+7 A d-11 B d) \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+6 d (-4 B c-2 A d+3 B d) \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 B d^2 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-6 d (-4 B c-2 A d+3 B d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 B d^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{6 f (a (1+\sin (e+f x)))^{3/2}} \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. \(693\) vs.
\(2(180)=360\).
time = 7.08, size = 694, normalized size = 3.42
method | result | size |
default | \(-\frac {\left (\sin \left (f x +e \right ) \left (3 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2}+18 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c d -21 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{2}+24 A \,d^{2} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}+9 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2}-42 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c d +33 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{2}-8 B \,d^{2} \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}+48 B c d \,a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}-24 B \,d^{2} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}\right )+3 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2}+18 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c d -21 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{2}+6 A \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{2}-12 A \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c d +30 A \,d^{2} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}+9 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2}-42 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c d +33 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{2}-8 B \,d^{2} \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}-6 B \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{2}+60 B c d \,a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}-30 B \,d^{2} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(694\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 610 vs.
\(2 (189) = 378\).
time = 0.46, size = 610, normalized size = 3.00 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (2 \, {\left (A + 3 \, B\right )} c^{2} + 4 \, {\left (3 \, A - 7 \, B\right )} c d - 2 \, {\left (7 \, A - 11 \, B\right )} d^{2} - {\left ({\left (A + 3 \, B\right )} c^{2} + 2 \, {\left (3 \, A - 7 \, B\right )} c d - {\left (7 \, A - 11 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left ({\left (A + 3 \, B\right )} c^{2} + 2 \, {\left (3 \, A - 7 \, B\right )} c d - {\left (7 \, A - 11 \, B\right )} d^{2}\right )} \cos \left (f x + e\right ) + {\left (2 \, {\left (A + 3 \, B\right )} c^{2} + 4 \, {\left (3 \, A - 7 \, B\right )} c d - 2 \, {\left (7 \, A - 11 \, B\right )} d^{2} + {\left ({\left (A + 3 \, B\right )} c^{2} + 2 \, {\left (3 \, A - 7 \, B\right )} c d - {\left (7 \, A - 11 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (4 \, B d^{2} \cos \left (f x + e\right )^{3} - 3 \, {\left (A - B\right )} c^{2} + 6 \, {\left (A - B\right )} c d - 3 \, {\left (A - B\right )} d^{2} - 4 \, {\left (6 \, B c d + {\left (3 \, A - 4 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left ({\left (A - B\right )} c^{2} - 2 \, {\left (A - 5 \, B\right )} c d + 5 \, {\left (A - B\right )} d^{2}\right )} \cos \left (f x + e\right ) - {\left (4 \, B d^{2} \cos \left (f x + e\right )^{2} - 3 \, {\left (A - B\right )} c^{2} + 6 \, {\left (A - B\right )} c d - 3 \, {\left (A - B\right )} d^{2} + 12 \, {\left (2 \, B c d + {\left (A - B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{24 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \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 {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{2}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 478 vs.
\(2 (189) = 378\).
time = 0.61, size = 478, normalized size = 2.35 \begin {gather*} \frac {\frac {3 \, \sqrt {2} {\left (A \sqrt {a} c^{2} + 3 \, B \sqrt {a} c^{2} + 6 \, A \sqrt {a} c d - 14 \, B \sqrt {a} c d - 7 \, A \sqrt {a} d^{2} + 11 \, B \sqrt {a} d^{2}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (A \sqrt {a} c^{2} + 3 \, B \sqrt {a} c^{2} + 6 \, A \sqrt {a} c d - 14 \, B \sqrt {a} c d - 7 \, A \sqrt {a} d^{2} + 11 \, B \sqrt {a} d^{2}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (A \sqrt {a} c^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} c^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, A \sqrt {a} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B \sqrt {a} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + A \sqrt {a} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, B a^{\frac {9}{2}} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24 \, f} \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 {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2}{{\left (a+a\,\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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