Optimal. Leaf size=145 \[ -\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a) (c+d \sec (e+f x))}-\frac {2 d (2 c+d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a f (c-d)^{5/2} (c+d)^{3/2}}+\frac {(c+2 d) \tan (e+f x)}{f (c-d)^2 (c+d) (a \sec (e+f x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3987, 103, 152, 12, 93, 205} \[ -\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a) (c+d \sec (e+f x))}+\frac {2 d (2 c+d) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{f (c-d)^{5/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c+2 d) \tan (e+f x)}{f (c-d)^2 (c+d) (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 93
Rule 103
Rule 152
Rule 205
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(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 {1}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x)) (c+d \sec (e+f x))}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {a^2 (c+d)-a^2 d x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+2 d) \tan (e+f x)}{(c-d)^2 (c+d) f (a+a \sec (e+f x))}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x)) (c+d \sec (e+f x))}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {a^4 d (2 c+d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+2 d) \tan (e+f x)}{(c-d)^2 (c+d) f (a+a \sec (e+f x))}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x)) (c+d \sec (e+f x))}+\frac {(a d (2 c+d) \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{(c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+2 d) \tan (e+f x)}{(c-d)^2 (c+d) f (a+a \sec (e+f x))}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x)) (c+d \sec (e+f x))}+\frac {(2 a d (2 c+d) \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {a-a \sec (e+f x)}}\right )}{(c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+2 d) \tan (e+f x)}{(c-d)^2 (c+d) f (a+a \sec (e+f x))}+\frac {2 d (2 c+d) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right ) \tan (e+f x)}{(c-d)^{5/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x)) (c+d \sec (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.25, size = 286, normalized size = 1.97 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) (c \cos (e+f x)+d) \left (\frac {2 d (2 c+d) (\sin (e)+i \cos (e)) \cos \left (\frac {1}{2} (e+f x)\right ) (c \cos (e+f x)+d) \tan ^{-1}\left (\frac {(\sin (e)+i \cos (e)) \left (\tan \left (\frac {f x}{2}\right ) (c \cos (e)-d)+c \sin (e)\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right )}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {d^2 \cos \left (\frac {1}{2} (e+f x)\right ) (c \sin (f x)-d \sin (e))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right )}+\sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) (c \cos (e+f x)+d)\right )}{a f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 691, normalized size = 4.77 \[ \left [\frac {{\left (2 \, c d^{2} + d^{3} + {\left (2 \, c^{2} d + c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4} + {\left (c^{4} + c^{3} d - c d^{3} - d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a c^{6} - a c^{5} d - 2 \, a c^{4} d^{2} + 2 \, a c^{3} d^{3} + a c^{2} d^{4} - a c d^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right )} f \cos \left (f x + e\right ) + {\left (a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right )} f\right )}}, -\frac {{\left (2 \, c d^{2} + d^{3} + {\left (2 \, c^{2} d + c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) - {\left (c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4} + {\left (c^{4} + c^{3} d - c d^{3} - d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (a c^{6} - a c^{5} d - 2 \, a c^{4} d^{2} + 2 \, a c^{3} d^{3} + a c^{2} d^{4} - a c d^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right )} f \cos \left (f x + e\right ) + {\left (a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 228, normalized size = 1.57 \[ -\frac {\frac {2 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}} - \frac {2 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )} {\left (2 \, c d + d^{2}\right )}}{{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a c^{2} - 2 \, a c d + a d^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.79, size = 146, normalized size = 1.01 \[ \frac {\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{c^{2}-2 c d +d^{2}}+\frac {4 d \left (-\frac {d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 \left (c +d \right ) \left (\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c -\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d -c -d \right )}-\frac {\left (2 c +d \right ) \arctanh \left (\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c -d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{2}}}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.04, size = 187, normalized size = 1.29 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f\,{\left (c-d\right )}^2}-\frac {2\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a\,c^3-3\,a\,c^2\,d+3\,a\,c\,d^2-a\,d^3\right )-a\,d^3-a\,c^3+a\,c\,d^2+a\,c^2\,d\right )}-\frac {2\,d\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-2\,d\right )\,\left (a\,c^2-2\,a\,c\,d+a\,d^2\right )}{2\,a\,\sqrt {c+d}\,{\left (c-d\right )}^{5/2}}\right )\,\left (2\,c+d\right )}{a\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________