Optimal. Leaf size=428 \[ \frac {3 a^2 b F_1\left (\frac {1}{2};-\frac {n p}{2},3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}+\frac {b^3 F_1\left (\frac {1}{2};\frac {1}{2} (-2-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}-\frac {3 a b^2 F_1\left (\frac {1}{2};\frac {1}{2} (-1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1-n p)} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}-\frac {a^3 F_1\left (\frac {1}{2};\frac {1}{2} (1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cot (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (1-n p)} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f} \]
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Rubi [A]
time = 0.46, antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2905, 2903,
3268, 440, 16} \begin {gather*} \frac {3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n F_1\left (\frac {1}{2};-\frac {n p}{2},3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )^3}-\frac {3 a b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-n p-1)} \left (c (d \sin (e+f x))^p\right )^n F_1\left (\frac {1}{2};\frac {1}{2} (-n p-1),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )^3}+\frac {b^3 \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n F_1\left (\frac {1}{2};\frac {1}{2} (-n p-2),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )^3}-\frac {a^3 \cot (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (1-n p)} \left (c (d \sin (e+f x))^p\right )^n F_1\left (\frac {1}{2};\frac {1}{2} (1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 440
Rule 2903
Rule 2905
Rule 3268
Rubi steps
\begin {align*} \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(a+b \sin (e+f x))^3} \, dx &=\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p}}{(a+b \sin (e+f x))^3} \, dx\\ &=\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \left (\frac {a^3 (d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3}-\frac {3 a^2 b \sin (e+f x) (d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3}+\frac {3 a b^2 \sin ^2(e+f x) (d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3}+\frac {b^3 \sin ^3(e+f x) (d \sin (e+f x))^{n p}}{\left (-a^2+b^2 \sin ^2(e+f x)\right )^3}\right ) \, dx\\ &=\left (a^3 (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3} \, dx-\left (3 a^2 b (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {\sin (e+f x) (d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3} \, dx+\left (3 a b^2 (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {\sin ^2(e+f x) (d \sin (e+f x))^{n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3} \, dx+\left (b^3 (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {\sin ^3(e+f x) (d \sin (e+f x))^{n p}}{\left (-a^2+b^2 \sin ^2(e+f x)\right )^3} \, dx\\ &=\frac {\left (b^3 (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{3+n p}}{\left (-a^2+b^2 \sin ^2(e+f x)\right )^3} \, dx}{d^3}+\frac {\left (3 a b^2 (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{2+n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3} \, dx}{d^2}-\frac {\left (3 a^2 b (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{1+n p}}{\left (a^2-b^2 \sin ^2(e+f x)\right )^3} \, dx}{d}-\frac {\left (a^3 d (d \sin (e+f x))^{-n p+2 \left (-\frac {1}{2}+\frac {n p}{2}\right )} \sin ^2(e+f x)^{\frac {1}{2}-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1}{2} (-1+n p)}}{\left (a^2-b^2+b^2 x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {a^3 F_1\left (\frac {1}{2};\frac {1}{2} (1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cot (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (1-n p)} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}+\frac {\left (3 a^2 b \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {n p}{2}}}{\left (a^2-b^2+b^2 x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}-\frac {\left (b^3 \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1}{2} (2+n p)}}{\left (-a^2+b^2-b^2 x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}-\frac {\left (3 a b^2 (d \sin (e+f x))^{-n p+2 \left (\frac {1}{2}+\frac {n p}{2}\right )} \sin ^2(e+f x)^{-\frac {1}{2}-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1}{2} (1+n p)}}{\left (a^2-b^2+b^2 x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{d f}\\ &=\frac {3 a^2 b F_1\left (\frac {1}{2};-\frac {n p}{2},3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}+\frac {b^3 F_1\left (\frac {1}{2};\frac {1}{2} (-2-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin ^2(e+f x)^{-\frac {n p}{2}} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}-\frac {3 a b^2 F_1\left (\frac {1}{2};\frac {1}{2} (-1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \sin (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1-n p)} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}-\frac {a^3 F_1\left (\frac {1}{2};\frac {1}{2} (1-n p),3;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cot (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (1-n p)} \left (c (d \sin (e+f x))^p\right )^n}{\left (a^2-b^2\right )^3 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2570\) vs. \(2(428)=856\).
time = 15.95, size = 2570, normalized size = 6.00 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.58, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{\left (a +b \sin \left (f x +e \right )\right )^{3}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\left (a + b \sin {\left (e + f x \right )}\right )^{3}}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{{\left (a+b\,\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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