Optimal. Leaf size=42 \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {3950} \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3950
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.48, size = 63, normalized size = 1.50 \[ \frac {a \tan \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \sec (e+f x)}}{c^3 f (\sec (e+f x)-1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 95, normalized size = 2.26 \[ \frac {a \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )} \]
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 = 1.94, size = 73, normalized size = 1.74 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (\sin ^{3}\left (f x +e \right )\right ) a}{4 f \left (-1+\cos \left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \cos \left (f x +e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.98, size = 533, normalized size = 12.69 \[ \frac {2 \, {\left (6 \, a \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 6 \, a \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, a \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - {\left (a \sin \left (3 \, f x + 3 \, e\right ) + a \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (a \cos \left (3 \, f x + 3 \, e\right ) + a \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - {\left (6 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} \sin \left (3 \, f x + 3 \, e\right ) - a \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c^{3} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \sin \left (2 \, f x + 2 \, e\right )^{2} - 48 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, c^{3} \sin \left (f x + e\right )^{2} - 8 \, c^{3} \cos \left (f x + e\right ) + c^{3} - 2 \, {\left (4 \, c^{3} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 8 \, {\left (6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 12 \, {\left (4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (2 \, c^{3} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 16 \, {\left (3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.83, size = 165, normalized size = 3.93 \[ -\frac {2\,a\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (6\,\sin \left (e+f\,x\right )-8\,\sin \left (2\,e+2\,f\,x\right )+7\,\sin \left (3\,e+3\,f\,x\right )-4\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (5\,e+5\,f\,x\right )\right )}{c^3\,f\,\left (48\,\cos \left (e+f\,x\right )+15\,\cos \left (2\,e+2\,f\,x\right )-40\,\cos \left (3\,e+3\,f\,x\right )+26\,\cos \left (4\,e+4\,f\,x\right )-8\,\cos \left (5\,e+5\,f\,x\right )+\cos \left (6\,e+6\,f\,x\right )-42\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________