Optimal. Leaf size=50 \[ \frac {\, _2F_1\left (3,\frac {5+n}{2};\frac {7+n}{2};-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{5+n}}{b^5 f (5+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2671, 371}
\begin {gather*} \frac {(b \tan (e+f x))^{n+5} \, _2F_1\left (3,\frac {n+5}{2};\frac {n+7}{2};-\tan ^2(e+f x)\right )}{b^5 f (n+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 2671
Rubi steps
\begin {align*} \int \sin ^4(e+f x) (b \tan (e+f x))^n \, dx &=\frac {b \text {Subst}\left (\int \frac {x^{4+n}}{\left (b^2+x^2\right )^3} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (3,\frac {5+n}{2};\frac {7+n}{2};-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{5+n}}{b^5 f (5+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 4.99, size = 916, normalized size = 18.32 \begin {gather*} \frac {64 (3+n) \left (F_1\left (\frac {1+n}{2};n,3;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 F_1\left (\frac {1+n}{2};n,4;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+F_1\left (\frac {1+n}{2};n,5;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sin ^5\left (\frac {1}{2} (e+f x)\right ) (b \tan (e+f x))^n}{f (1+n) \left ((3+n) F_1\left (\frac {1+n}{2};n,3;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))+(3+n) F_1\left (\frac {1+n}{2};n,5;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))+2 \left (-5 F_1\left (\frac {3+n}{2};n,6;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n F_1\left (\frac {3+n}{2};1+n,3;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 n F_1\left (\frac {3+n}{2};1+n,4;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n F_1\left (\frac {3+n}{2};1+n,5;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-6 F_1\left (\frac {1+n}{2};n,4;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 n F_1\left (\frac {1+n}{2};n,4;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+3 F_1\left (\frac {3+n}{2};n,4;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (-1+\cos (e+f x))-8 F_1\left (\frac {3+n}{2};n,5;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (-1+\cos (e+f x))+5 F_1\left (\frac {3+n}{2};n,6;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)-n F_1\left (\frac {3+n}{2};1+n,3;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)+2 n F_1\left (\frac {3+n}{2};1+n,4;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)-n F_1\left (\frac {3+n}{2};1+n,5;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.99, size = 0, normalized size = 0.00 \[\int \left (\sin ^{4}\left (f x +e \right )\right ) \left (b \tan \left (f x +e \right )\right )^{n}\, dx\]
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 = 0.00, 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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \tan {\left (e + f x \right )}\right )^{n} \sin ^{4}{\left (e + f x \right )}\, dx \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 = -0.02 \begin {gather*} \int {\sin \left (e+f\,x\right )}^4\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________