Optimal. Leaf size=123 \[ \frac {2 (5 a+2 b) \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac {\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^3 f}-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{5 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4134, 462, 453, 264} \[ -\frac {\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac {2 (5 a+2 b) \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{5 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 453
Rule 462
Rule 4134
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{5 a f}+\frac {\operatorname {Subst}\left (\int \frac {-2 (5 a+2 b)+5 a x^2}{x^4 \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{5 a f}\\ &=\frac {2 (5 a+2 b) \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{5 a f}+\frac {\left (15 a^2+20 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{15 a^2 f}\\ &=-\frac {\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac {2 (5 a+2 b) \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)}}{5 a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.93, size = 93, normalized size = 0.76 \[ -\frac {\sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left (3 a^2 \cos (4 (e+f x))+89 a^2-4 a (7 a+4 b) \cos (2 (e+f x))+144 a b+64 b^2\right )}{240 a^3 f \sqrt {a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 87, normalized size = 0.71 \[ -\frac {{\left (3 \, a^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (5 \, a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{2} + 20 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.96, size = 105, normalized size = 0.85 \[ -\frac {\left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \left (3 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2}-10 a^{2} \left (\cos ^{2}\left (f x +e \right )\right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) a b +15 a^{2}+20 a b +8 b^{2}\right )}{15 f \sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right ) a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 162, normalized size = 1.32 \[ -\frac {\frac {15 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a} - \frac {10 \, {\left ({\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 3 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )\right )}}{a^{2}} + \frac {3 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{5} - 10 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} b \cos \left (f x + e\right )^{3} + 15 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b^{2} \cos \left (f x + e\right )}{a^{3}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (e+f\,x\right )}^5}{\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}} \,d x \]
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]
________________________________________________________________________________________