Optimal. Leaf size=226 \[ \frac {\, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f}-\frac {2 \, _2F_1\left (\frac {3}{n},-p;\frac {3+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{3 f}+\frac {\, _2F_1\left (\frac {5}{n},-p;\frac {5+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^5(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 226, normalized size of antiderivative = 1.00, 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 = {3302, 1907,
252, 251, 372, 371} \begin {gather*} \frac {\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{f}+\frac {\sin ^5(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{n},-p;\frac {n+5}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{5 f}-\frac {2 \sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 252
Rule 371
Rule 372
Rule 1907
Rule 3302
Rubi steps
\begin {align*} \int \cos ^5(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right )^2 \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\left (a+b x^n\right )^p-2 x^2 \left (a+b x^n\right )^p+x^4 \left (a+b x^n\right )^p\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}+\frac {\text {Subst}\left (\int x^4 \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}-\frac {2 \text {Subst}\left (\int x^2 \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \left (1+\frac {b x^n}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}+\frac {\left (\left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int x^4 \left (1+\frac {b x^n}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}-\frac {\left (2 \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int x^2 \left (1+\frac {b x^n}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f}-\frac {2 \, _2F_1\left (\frac {3}{n},-p;\frac {3+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{3 f}+\frac {\, _2F_1\left (\frac {5}{n},-p;\frac {5+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^5(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{5 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 155, normalized size = 0.69 \begin {gather*} \frac {\sin (e+f x) \left (15 \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right )-10 \, _2F_1\left (\frac {3}{n},-p;\frac {3+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^2(e+f x)+3 \, _2F_1\left (\frac {5}{n},-p;\frac {5+n}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^4(e+f x)\right ) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{15 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \left (\cos ^{5}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.45, size = 25, normalized size = 0.11 \begin {gather*} {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )^{5}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (e+f\,x\right )}^5\,{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________