Optimal. Leaf size=148 \[ \frac {2^{\frac {1}{2}+m} (A (2-m)-B (3+m)) \, _2F_1\left (-\frac {5}{2},\frac {1}{2}-m;-\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{2+m}}{5 a^2 c^3 f (2-m)}+\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)} \]
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Rubi [A]
time = 0.22, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2939,
2768, 72, 71} \begin {gather*} \frac {B \sec ^5(e+f x) (a \sin (e+f x)+a)^{m+3}}{a^3 c^3 f (2-m)}+\frac {2^{m+\frac {1}{2}} (A (2-m)-B (m+3)) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {1}{2}-m;-\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{5 a^2 c^3 f (2-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rule 2939
Rule 3046
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx &=\frac {\int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} (A+B \sin (e+f x)) \, dx}{a^3 c^3}\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (A-\frac {B (3+m)}{2-m}\right ) \int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} \, dx}{a^3 c^3}\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (\left (A-\frac {B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f}\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (2^{-\frac {1}{2}+m} \left (A-\frac {B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f}\\ &=\frac {2^{\frac {1}{2}+m} \left (A-\frac {B (3+m)}{2-m}\right ) \, _2F_1\left (-\frac {5}{2},\frac {1}{2}-m;-\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{2+m}}{5 a^2 c^3 f}+\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 6.82, size = 12302, normalized size = 83.12 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 2.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{3}}\, dx\]
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(-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 {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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