Optimal. Leaf size=137 \[ \frac {c F_1\left (\frac {1+n}{2};\frac {1-m}{2},-p;\frac {3+n}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) (c \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac {1-m}{2}} (d \sin (e+f x))^{1+n} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{d f (1+n)} \]
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Rubi [A]
time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3281, 525, 524}
\begin {gather*} \frac {c \cos ^2(e+f x)^{\frac {1-m}{2}} (c \cos (e+f x))^{m-1} (d \sin (e+f x))^{n+1} \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {n+1}{2};\frac {1-m}{2},-p;\frac {n+3}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right )}{d f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3281
Rubi steps
\begin {align*} \int (c \cos (e+f x))^m (d \sin (e+f x))^n \left (a+b \sin ^2(e+f x)\right )^p \, dx &=\frac {\left (c (c \cos (e+f x))^{2 \left (-\frac {1}{2}+\frac {m}{2}\right )} \cos ^2(e+f x)^{\frac {1}{2}-\frac {m}{2}}\right ) \text {Subst}\left (\int (d x)^n \left (1-x^2\right )^{\frac {1}{2} (-1+m)} \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (c (c \cos (e+f x))^{2 \left (-\frac {1}{2}+\frac {m}{2}\right )} \cos ^2(e+f x)^{\frac {1}{2}-\frac {m}{2}} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int (d x)^n \left (1-x^2\right )^{\frac {1}{2} (-1+m)} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {c F_1\left (\frac {1+n}{2};\frac {1-m}{2},-p;\frac {3+n}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) (c \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac {1-m}{2}} (d \sin (e+f x))^{1+n} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{d f (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 135, normalized size = 0.99 \begin {gather*} \frac {F_1\left (\frac {1+n}{2};\frac {1-m}{2},-p;\frac {3+n}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) (c \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1-m}{2}} (d \sin (e+f x))^n \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x)}{f (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.01, size = 0, normalized size = 0.00 \[\int \left (\cos \left (f x +e \right ) c \right )^{m} \left (d \sin \left (f x +e \right )\right )^{n} \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 = 1.21, size = 39, normalized size = 0.28 \begin {gather*} {\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \left (c \cos \left (f x + e\right )\right )^{m} \left (d \sin \left (f x + e\right )\right )^{n}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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