Optimal. Leaf size=80 \[ -\frac {b^5 (b \tan (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2591, 270} \[ -\frac {b^5 (b \tan (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2591
Rubi steps
\begin {align*} \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx &=\frac {b \operatorname {Subst}\left (\int x^{-6+n} \left (b^2+x^2\right )^2 \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {b \operatorname {Subst}\left (\int \left (b^4 x^{-6+n}+2 b^2 x^{-4+n}+x^{-2+n}\right ) \, dx,x,b \tan (e+f x)\right )}{f}\\ &=-\frac {b^5 (b \tan (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 69, normalized size = 0.86 \[ \frac {b \csc ^4(e+f x) \left (2 (n-3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2-6 n+8\right ) (b \tan (e+f x))^{n-1}}{f (n-5) (n-3) (n-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 144, normalized size = 1.80 \[ \frac {{\left (8 \, \cos \left (f x + e\right )^{5} + 4 \, {\left (n - 5\right )} \cos \left (f x + e\right )^{3} + {\left (n^{2} - 8 \, n + 15\right )} \cos \left (f x + e\right )\right )} \left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n}}{{\left ({\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{4} + f n^{3} - 9 \, f n^{2} - 2 \, {\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{2} + 23 \, f n - 15 \, f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.01, size = 26124, normalized size = 326.55 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 81, normalized size = 1.01 \[ \frac {\frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 1\right )} \tan \left (f x + e\right )} + \frac {2 \, b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 3\right )} \tan \left (f x + e\right )^{3}} + \frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 5\right )} \tan \left (f x + e\right )^{5}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________