Optimal. Leaf size=133 \[ \frac {d^3 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {(c-d) \tan (e+f x) \left (\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)+2 \left (2 c^2+8 c d+11 d^2\right )\right )}{15 a f (a \sec (e+f x)+a)^2}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 193, normalized size of antiderivative = 1.45, 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, 98, 145, 63, 217, 203} \[ \frac {2 d^3 \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d) \tan (e+f x) \left (\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)+2 \left (2 c^2+8 c d+11 d^2\right )\right )}{15 a f (a \sec (e+f x)+a)^2}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 98
Rule 145
Rule 203
Rule 217
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^3}{\sqrt {a-a x} (a+a x)^{7/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+d \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(c+d x) \left (-a^2 \left (2 c^2+5 c d-2 d^2\right )-5 a^2 d^2 x\right )}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+d \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(c-d) \left (2 \left (2 c^2+8 c d+11 d^2\right )+\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}-\frac {\left (d^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+d \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(c-d) \left (2 \left (2 c^2+8 c d+11 d^2\right )+\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {\left (2 d^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d) (c+d \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(c-d) \left (2 \left (2 c^2+8 c d+11 d^2\right )+\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {\left (2 d^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 d^3 \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(c-d) (c+d \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(c-d) \left (2 \left (2 c^2+8 c d+11 d^2\right )+\left (2 c^2+11 c d+29 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.48, size = 295, normalized size = 2.22 \[ \frac {(c-d) \sec \left (\frac {e}{2}\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \left (-15 \left (2 c^2+5 c d+5 d^2\right ) \sin \left (e+\frac {f x}{2}\right )+5 \left (8 c^2+17 c d+29 d^2\right ) \sin \left (\frac {f x}{2}\right )+20 c^2 \sin \left (e+\frac {3 f x}{2}\right )-15 c^2 \sin \left (2 e+\frac {3 f x}{2}\right )+7 c^2 \sin \left (2 e+\frac {5 f x}{2}\right )+65 c d \sin \left (e+\frac {3 f x}{2}\right )-15 c d \sin \left (2 e+\frac {3 f x}{2}\right )+16 c d \sin \left (2 e+\frac {5 f x}{2}\right )+95 d^2 \sin \left (e+\frac {3 f x}{2}\right )-15 d^2 \sin \left (2 e+\frac {3 f x}{2}\right )+22 d^2 \sin \left (2 e+\frac {5 f x}{2}\right )\right )-240 d^3 \cos ^6\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{30 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 248, normalized size = 1.86 \[ \frac {15 \, {\left (d^{3} \cos \left (f x + e\right )^{3} + 3 \, d^{3} \cos \left (f x + e\right )^{2} + 3 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (d^{3} \cos \left (f x + e\right )^{3} + 3 \, d^{3} \cos \left (f x + e\right )^{2} + 3 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, c^{3} + 9 \, c^{2} d + 21 \, c d^{2} - 32 \, d^{3} + {\left (7 \, c^{3} + 9 \, c^{2} d + 6 \, c d^{2} - 22 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, c^{3} + 9 \, c^{2} d + 6 \, c d^{2} - 17 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
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 = 0.84, size = 286, normalized size = 2.15 \[ \frac {3 c^{2} \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d}{4 f \,a^{3}}+\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c \,d^{2}}{2 f \,a^{3}}+\frac {3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c \,d^{2}}{4 f \,a^{3}}-\frac {3 \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{2} d}{20 f \,a^{3}}+\frac {3 \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c \,d^{2}}{20 f \,a^{3}}-\frac {7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d^{3}}{4 f \,a^{3}}-\frac {d^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f \,a^{3}}-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{3}}{6 f \,a^{3}}-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{3}}{3 f \,a^{3}}+\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{3}}{4 f \,a^{3}}+\frac {d^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f \,a^{3}}+\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{3}}{20 f \,a^{3}}-\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{3}}{20 f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 307, normalized size = 2.31 \[ -\frac {d^{3} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} - \frac {3 \, c d^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {c^{3} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {9 \, c^{2} d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.80, size = 147, normalized size = 1.11 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {{\left (c-d\right )}^3}{4\,a^3}-\frac {3\,\left (c+d\right )\,{\left (c-d\right )}^2}{4\,a^3}+\frac {3\,{\left (c+d\right )}^2\,\left (c-d\right )}{4\,a^3}\right )}{f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {{\left (c-d\right )}^3}{12\,a^3}-\frac {\left (c+d\right )\,{\left (c-d\right )}^2}{4\,a^3}\right )}{f}+\frac {2\,d^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\left (c-d\right )}^3}{20\,a^3\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c^{3} \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________