Optimal. Leaf size=65 \[ -\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2702, 276}
\begin {gather*} -\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 276
Rule 2702
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx &=\frac {b^5 \text {Subst}\left (\int \frac {\left (-1+\frac {x^2}{b^2}\right )^2}{x^{13/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {b^5 \text {Subst}\left (\int \left (\frac {1}{x^{13/2}}-\frac {2}{b^2 x^{9/2}}+\frac {1}{b^4 x^{5/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=-\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 42, normalized size = 0.65 \begin {gather*} \frac {b (-415+180 \cos (2 (e+f x))-21 \cos (4 (e+f x)))}{924 f (b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.21, size = 46, normalized size = 0.71
method | result | size |
default | \(-\frac {2 \left (21 \left (\cos ^{4}\left (f x +e \right )\right )-66 \left (\cos ^{2}\left (f x +e \right )\right )+77\right ) \cos \left (f x +e \right )}{231 f \sqrt {\frac {b}{\cos \left (f x +e \right )}}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 53, normalized size = 0.82 \begin {gather*} -\frac {2 \, {\left (21 \, b^{4} - \frac {66 \, b^{4}}{\cos \left (f x + e\right )^{2}} + \frac {77 \, b^{4}}{\cos \left (f x + e\right )^{4}}\right )} b}{231 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 55, normalized size = 0.85 \begin {gather*} -\frac {2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 66 \, \cos \left (f x + e\right )^{4} + 77 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{231 \, b f} \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 [A]
time = 3.66, size = 92, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (21 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 66 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3} + 77 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )\right )}}{231 \, b^{6} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \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 \frac {{\sin \left (e+f\,x\right )}^5}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________