Optimal. Leaf size=144 \[ \frac {2 a^{3/2} c^2 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a d (2 c+d) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}-\frac {2 d^2 \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt {a \sec (e+f x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3940, 88, 63, 206} \[ \frac {2 a^{3/2} c^2 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a d (2 c+d) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}-\frac {2 d^2 \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt {a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 88
Rule 206
Rule 3940
Rubi steps
\begin {align*} \int \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^2}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \left (\frac {d (2 c+d)}{\sqrt {a-a x}}+\frac {c^2}{x \sqrt {a-a x}}-\frac {d^2 \sqrt {a-a x}}{a}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^2 c^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 a c^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^{3/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.59, size = 444, normalized size = 3.08 \[ \frac {\sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )} \csc ^3\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sec (e+f x)+1)} (c+d \sec (e+f x))^2 \left (256 \sin ^6\left (\frac {1}{2} (e+f x)\right ) \left (-2 c \sin ^2\left (\frac {1}{2} (e+f x)\right )+c+d\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {7}{2};1,\frac {9}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )+1024 \sin ^6\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (c^2 \left (2 \sin ^4\left (\frac {1}{2} (e+f x)\right )-3 \sin ^2\left (\frac {1}{2} (e+f x)\right )+1\right )+c d \left (2-3 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )+d^2\right )-\frac {7 \sqrt {2} \left (\sqrt {2} \sqrt {\sin ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )} \left (4 \sin ^2\left (\frac {1}{2} (e+f x)\right )+3\right )-3 \sin ^{-1}\left (\sqrt {2} \sqrt {\sin ^2\left (\frac {1}{2} (e+f x)\right )}\right )\right ) \left (c^2 \left (12 \sin ^4\left (\frac {1}{2} (e+f x)\right )-20 \sin ^2\left (\frac {1}{2} (e+f x)\right )+15\right )+10 c d \left (3-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )+15 d^2\right )}{\sqrt {\sin ^2\left (\frac {1}{2} (e+f x)\right )}}\right )}{672 f \sec ^{\frac {5}{2}}(e+f x) (c \cos (e+f x)+d)^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 320, normalized size = 2.22 \[ \left [\frac {3 \, {\left (c^{2} \cos \left (f x + e\right )^{2} + c^{2} \cos \left (f x + e\right )\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) + 2 \, {\left (d^{2} + 2 \, {\left (3 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{3 \, {\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\right )}}, -\frac {2 \, {\left (3 \, {\left (c^{2} \cos \left (f x + e\right )^{2} + c^{2} \cos \left (f x + e\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - {\left (d^{2} + 2 \, {\left (3 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{3 \, {\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\right )}}\right ] \]
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 [A] time = 1.74, size = 248, normalized size = 1.72 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (3 \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \cos \left (f x +e \right ) \sqrt {2}\, c^{2}+3 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, c^{2} \sin \left (f x +e \right )-24 \left (\cos ^{2}\left (f x +e \right )\right ) c d -8 \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+24 \cos \left (f x +e \right ) c d +4 \cos \left (f x +e \right ) d^{2}+4 d^{2}\right )}{6 f \sin \left (f x +e \right ) \cos \left (f x +e \right )} \]
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 \[ \int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (c + d \sec {\left (e + f x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________