Optimal. Leaf size=197 \[ -\frac{2 \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right )}{3 f}-\frac{(a-b (4 p+5)) \sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right )}{b f (4 p+5)}+\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{p+1}}{b f (4 p+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.221972, antiderivative size = 191, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3223, 1207, 1204, 246, 245, 365, 364} \[ -\frac{2 \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right )}{3 f}+\frac{\left (1-\frac{a}{4 b p+5 b}\right ) \sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right )}{f}+\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{p+1}}{b f (4 p+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3223
Rule 1207
Rule 1204
Rule 246
Rule 245
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \cos ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^2 \left (a+b x^4\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{1+p}}{b f (5+4 p)}+\frac{\operatorname{Subst}\left (\int \left (-a+b (5+4 p)-2 b (5+4 p) x^2\right ) \left (a+b x^4\right )^p \, dx,x,\sin (e+f x)\right )}{b f (5+4 p)}\\ &=\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{1+p}}{b f (5+4 p)}+\frac{\operatorname{Subst}\left (\int \left (-a \left (1-\frac{b (5+4 p)}{a}\right ) \left (a+b x^4\right )^p-2 b (5+4 p) x^2 \left (a+b x^4\right )^p\right ) \, dx,x,\sin (e+f x)\right )}{b f (5+4 p)}\\ &=\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{1+p}}{b f (5+4 p)}-\frac{2 \operatorname{Subst}\left (\int x^2 \left (a+b x^4\right )^p \, dx,x,\sin (e+f x)\right )}{f}-\frac{(a-5 b-4 b p) \operatorname{Subst}\left (\int \left (a+b x^4\right )^p \, dx,x,\sin (e+f x)\right )}{b f (5+4 p)}\\ &=\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{1+p}}{b f (5+4 p)}-\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 x^2 \left (1+\frac{b x^4}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}-\frac{\left ((a-5 b-4 b p) \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 \left (1+\frac{b x^4}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{b f (5+4 p)}\\ &=\frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^{1+p}}{b f (5+4 p)}-\frac{(a-b (5+4 p)) \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\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}}{b f (5+4 p)}-\frac{2 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\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}\\ \end{align*}
Mathematica [A] time = 0.147129, size = 141, normalized size = 0.72 \[ \frac{\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac{b \sin ^4(e+f x)}{a}+1\right )^{-p} \left (3 \sin ^4(e+f x) \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b \sin ^4(e+f x)}{a}\right )-10 \sin ^2(e+f x) \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right )+15 \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right )\right )}{15 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.021, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{5} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \cos \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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} \cos \left (f x + e\right )^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \cos \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]