Optimal. Leaf size=194 \[ -\frac {3 \left (c^2+6 c d+25 d^2\right ) (c-d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d^2 (c-9 d) \cos (e+f x)}{4 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {(3 c+13 d) (c-d)^2 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}} \]
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Rubi [A] time = 0.47, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2765, 2968, 3019, 2751, 2649, 206} \[ -\frac {3 \left (c^2+6 c d+25 d^2\right ) (c-d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d^2 (c-9 d) \cos (e+f x)}{4 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {(3 c+13 d) (c-d)^2 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2765
Rule 2968
Rule 3019
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-\frac {1}{2} a \left (3 c^2+9 c d-4 d^2\right )+\frac {1}{2} a (c-9 d) d \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {-\frac {1}{2} a c \left (3 c^2+9 c d-4 d^2\right )+\left (\frac {1}{2} a c (c-9 d) d-\frac {1}{2} a d \left (3 c^2+9 c d-4 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} a (c-9 d) d^2 \sin ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(c-d)^2 (3 c+13 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {\frac {1}{4} a^2 \left (3 c^3+15 c^2 d+53 c d^2-39 d^3\right )-a^2 (c-9 d) d^2 \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{8 a^4}\\ &=-\frac {(c-d)^2 (3 c+13 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {(c-9 d) d^2 \cos (e+f x)}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\left (3 (c-d) \left (c^2+6 c d+25 d^2\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2}\\ &=-\frac {(c-d)^2 (3 c+13 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {(c-9 d) d^2 \cos (e+f x)}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\left (3 (c-d) \left (c^2+6 c d+25 d^2\right )\right ) \operatorname {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 )}{16 a^2 f}\\ &=-\frac {3 (c-d) \left (c^2+6 c d+25 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d)^2 (3 c+13 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {(c-9 d) d^2 \cos (e+f x)}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a+a \sin (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.75, size = 400, normalized size = 2.06 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (11 c^3 \sin \left (\frac {1}{2} (e+f x)\right )-3 c^3 \sin \left (\frac {3}{2} (e+f x)\right )-11 c^3 \cos \left (\frac {1}{2} (e+f x)\right )-3 c^3 \cos \left (\frac {3}{2} (e+f x)\right )-9 c^2 d \sin \left (\frac {1}{2} (e+f x)\right )-15 c^2 d \sin \left (\frac {3}{2} (e+f x)\right )+9 c^2 d \cos \left (\frac {1}{2} (e+f x)\right )-15 c^2 d \cos \left (\frac {3}{2} (e+f x)\right )+(6+6 i) (-1)^{3/4} \left (c^3+5 c^2 d+19 c d^2-25 d^3\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )-15 c d^2 \sin \left (\frac {1}{2} (e+f x)\right )+39 c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+15 c d^2 \cos \left (\frac {1}{2} (e+f x)\right )+39 c d^2 \cos \left (\frac {3}{2} (e+f x)\right )+45 d^3 \sin \left (\frac {1}{2} (e+f x)\right )-69 d^3 \sin \left (\frac {3}{2} (e+f x)\right )-16 d^3 \sin \left (\frac {5}{2} (e+f x)\right )-45 d^3 \cos \left (\frac {1}{2} (e+f x)\right )-69 d^3 \cos \left (\frac {3}{2} (e+f x)\right )+16 d^3 \cos \left (\frac {5}{2} (e+f x)\right )\right )}{32 f (a (\sin (e+f x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 608, normalized size = 3.13 \[ -\frac {3 \, \sqrt {2} {\left ({\left (c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right )} \cos \left (f x + e\right )^{3} - 4 \, c^{3} - 20 \, c^{2} d - 76 \, c d^{2} + 100 \, d^{3} + 3 \, {\left (c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (4 \, c^{3} + 20 \, c^{2} d + 76 \, c d^{2} - 100 \, d^{3} - {\left (c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\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 (32 \, d^{3} \cos \left (f x + e\right )^{3} - 4 \, c^{3} + 12 \, c^{2} d - 12 \, c d^{2} + 4 \, d^{3} - {\left (3 \, c^{3} + 15 \, c^{2} d - 39 \, c d^{2} + 53 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (7 \, c^{3} + 3 \, c^{2} d - 27 \, c d^{2} + 81 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (32 \, d^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3} + 12 \, c^{2} d - 12 \, c d^{2} + 4 \, d^{3} + {\left (3 \, c^{3} + 15 \, c^{2} d - 39 \, c d^{2} + 85 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.32, size = 688, normalized size = 3.55 \[ -\frac {\left (2 \sin \left (f x +e \right ) \left (64 d^{3} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}+3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}+15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +57 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-75 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}\right )+\left (-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}-15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d -57 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}+75 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}-64 d^{3} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+6 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}+30 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +114 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-150 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}-6 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, c^{3}-30 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, c^{2} d +78 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, c \,d^{2}-42 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d^{3} \sqrt {a}+20 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{3}+36 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{2} d -132 c \,d^{2} a^{\frac {3}{2}} \sqrt {a -a \sin \left (f x +e \right )}+204 d^{3} 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 )}}{32 a^{\frac {9}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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