Optimal. Leaf size=265 \[ \frac {a^{5/2} (c-d) \left (A d (3 c+5 d)-B \left (5 c^2+5 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{7/2} f (c+d)^{3/2}}-\frac {a^3 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \cos (e+f x)}{3 d^3 f (c+d) \sqrt {a \sin (e+f x)+a}}-\frac {a^2 (-3 A d+5 B c+2 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 d^2 f (c+d)}+\frac {a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{d f (c+d) (c+d \sin (e+f x))} \]
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Rubi [A] time = 0.94, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2975, 2976, 2981, 2773, 208} \[ -\frac {a^3 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \cos (e+f x)}{3 d^3 f (c+d) \sqrt {a \sin (e+f x)+a}}+\frac {a^{5/2} (c-d) \left (A d (3 c+5 d)-B \left (5 c^2+5 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{7/2} f (c+d)^{3/2}}-\frac {a^2 (-3 A d+5 B c+2 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 d^2 f (c+d)}+\frac {a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{d f (c+d) (c+d \sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2773
Rule 2975
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx &=\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {(a+a \sin (e+f x))^{3/2} \left (-\frac {1}{2} a (3 B c-5 A d-2 B d)+\frac {1}{2} a (5 B c-3 A d+2 B d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac {a^2 (5 B c-3 A d+2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{d (c+d) f (c+d \sin (e+f x))}+\frac {2 \int \frac {\sqrt {a+a \sin (e+f x)} \left (-\frac {1}{4} a^2 \left (3 A (c-5 d) d-B \left (5 c^2-7 c d+6 d^2\right )\right )+\frac {1}{4} a^2 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{3 d^2 (c+d)}\\ &=-\frac {a^3 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \cos (e+f x)}{3 d^3 (c+d) f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 (5 B c-3 A d+2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (a^2 (c-d) \left (A d (3 c+5 d)-B \left (5 c^2+5 c d-2 d^2\right )\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)}\\ &=-\frac {a^3 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \cos (e+f x)}{3 d^3 (c+d) f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 (5 B c-3 A d+2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (a^3 (c-d) \left (A d (3 c+5 d)-B \left (5 c^2+5 c d-2 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d^3 (c+d) f}\\ &=\frac {a^{5/2} (c-d) \left (A d (3 c+5 d)-B \left (5 c^2+5 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{7/2} (c+d)^{3/2} f}-\frac {a^3 \left (3 A d (3 c+d)-B \left (15 c^2-5 c d-14 d^2\right )\right ) \cos (e+f x)}{3 d^3 (c+d) f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 (5 B c-3 A d+2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{d (c+d) f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 5.97, size = 460, normalized size = 1.74 \[ \frac {(a (\sin (e+f x)+1))^{5/2} \left (\frac {3 (c-d) \left (B \left (5 c^2+5 c d-2 d^2\right )-A d (3 c+5 d)\right ) \left (2 \log \left (\sqrt {d} \sqrt {c+d} \left (\tan ^2\left (\frac {1}{4} (e+f x)\right )+2 \tan \left (\frac {1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}-\frac {3 (c-d) \left (B \left (5 c^2+5 c d-2 d^2\right )-A d (3 c+5 d)\right ) \left (2 \log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+c+d\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}+12 \sqrt {d} (2 A d-4 B c+5 B d) \sin \left (\frac {1}{2} (e+f x)\right )-12 \sqrt {d} (2 A d-4 B c+5 B d) \cos \left (\frac {1}{2} (e+f x)\right )-\frac {12 \sqrt {d} (c-d)^2 (A d-B c) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))}-4 B d^{3/2} \sin \left (\frac {3}{2} (e+f x)\right )-4 B d^{3/2} \cos \left (\frac {3}{2} (e+f x)\right )\right )}{12 d^{7/2} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 2.59, size = 2046, normalized size = 7.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 2.54, size = 932, normalized size = 3.52 \[ \frac {a \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right ) d \left (9 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{2} d +6 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c \,d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} d^{3}-15 a^{2} \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) B \,c^{3}+21 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c \,d^{2}-6 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} d^{3}+2 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c d +2 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, d^{2}-6 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a c d -6 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,d^{2}+12 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{2}-6 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a c d -18 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,d^{2}\right )+9 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{3} d +6 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{2} d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c \,d^{3}+2 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c^{2} d +2 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c \,d^{2}-15 a^{2} \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) B \,c^{4}+21 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{2} d^{2}-6 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c \,d^{3}-9 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{2} d -3 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,d^{3}+15 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{3}-12 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{2} d -15 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a c \,d^{2}\right )}{3 d^{3} \left (c +d \right ) \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a \left (c +d \right ) d}\, \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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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