Optimal. Leaf size=239 \[ \frac{\sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{5}{4};2,-p;\frac{9}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{5 f}+\frac{2 \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{3}{4};2,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{3 f}+\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1}{4};2,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{f} \]
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Rubi [A] time = 0.216692, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3223, 1240, 430, 429, 511, 510} \[ \frac{\sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{5}{4};2,-p;\frac{9}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{5 f}+\frac{2 \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{3}{4};2,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{3 f}+\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1}{4};2,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 1240
Rule 430
Rule 429
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^p}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac{2 x^2 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac{x^4 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (\left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac{\left (\left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (1+\frac{b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac{\left (2 \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (1+\frac{b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{F_1\left (\frac{1}{4};2,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}}{f}+\frac{2 F_1\left (\frac{3}{4};2,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}}{3 f}+\frac{F_1\left (\frac{5}{4};2,-p;\frac{9}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right ) \sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac{b \sin ^4(e+f x)}{a}\right )^{-p}}{5 f}\\ \end{align*}
Mathematica [F] time = 8.72661, size = 0, normalized size = 0. \[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.212, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{3} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{4} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \sec \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \cos \left (f x + e\right )^{4} - 2 \, b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \sec \left (f x + e\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \sec \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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