Optimal. Leaf size=117 \[ -\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} c^{5/2} f}+\frac {5 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac {a^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 130, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3957, 3795, 203} \[ -\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} c^{5/2} f}+\frac {3 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 3795
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}-\frac {(3 a) \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{3/2}} \, dx}{4 c}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {3 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {\left (3 a^2\right ) \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{8 c^2}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {3 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{4 c^2 f}\\ &=-\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} c^{5/2} f}-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {3 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.63, size = 359, normalized size = 3.07 \[ -\frac {a^2 \csc \left (\frac {e}{2}\right ) e^{-\frac {1}{2} i (e+f x)} \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^3\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} (\sec (e+f x)+1)^2 \left (3 \sin \left (\frac {e}{2}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \sin ^4\left (\frac {1}{2} (e+f x)\right ) \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )+\frac {e^{-\frac {3 i e}{2}} \left (-1+e^{i e}\right ) \left (\cos \left (\frac {f x}{2}\right )+i \sin \left (\frac {f x}{2}\right )\right ) \left (-9 e^{i e} \sin \left (\frac {f x}{2}\right )+9 e^{2 i e} \sin \left (\frac {f x}{2}\right )-e^{3 i e} \sin \left (\frac {3 f x}{2}\right )-9 i e^{i e} \left (1+e^{i e}\right ) \cos \left (\frac {f x}{2}\right )+i \left (1+e^{3 i e}\right ) \cos \left (\frac {3 f x}{2}\right )+\sin \left (\frac {3 f x}{2}\right )\right )}{16 \sqrt {\sec (e+f x)}}\right )}{4 c^2 f (\sec (e+f x)-1)^2 \sqrt {c-c \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 429, normalized size = 3.67 \[ \left [-\frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 4 \, a^{2} \cos \left (f x + e\right )^{2} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{16 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}, \frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 4 \, a^{2} \cos \left (f x + e\right )^{2} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]
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 [B] time = 1.90, size = 230, normalized size = 1.97 \[ -\frac {a^{2} \left (-1+\cos \left (f x +e \right )\right )^{3} \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+3 \left (\cos ^{2}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )-4 \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-6 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )-5 \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+3 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )\right )}{f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{5} \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________