Optimal. Leaf size=97 \[ a^3 c^2 x+\frac {3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3966,
3855} \begin {gather*} \frac {3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {c^2 \tan ^3(e+f x) \left (3 a^3 \sec (e+f x)+4 a^3\right )}{12 f}-\frac {c^2 \tan (e+f x) \left (3 a^3 \sec (e+f x)+8 a^3\right )}{8 f}+a^3 c^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3855
Rule 3966
Rule 3989
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int (a+a \sec (e+f x)) \tan ^4(e+f x) \, dx\\ &=\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}-\frac {1}{4} \left (a^2 c^2\right ) \int (4 a+3 a \sec (e+f x)) \tan ^2(e+f x) \, dx\\ &=-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}+\frac {1}{8} \left (a^2 c^2\right ) \int (8 a+3 a \sec (e+f x)) \, dx\\ &=a^3 c^2 x-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}+\frac {1}{8} \left (3 a^3 c^2\right ) \int \sec (e+f x) \, dx\\ &=a^3 c^2 x+\frac {3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.86, size = 122, normalized size = 1.26 \begin {gather*} \frac {a^3 c^2 \sec ^4(e+f x) \left (72 e+72 f x+72 \tanh ^{-1}(\sin (e+f x)) \cos ^4(e+f x)+96 (e+f x) \cos (2 (e+f x))+24 e \cos (4 (e+f x))+24 f x \cos (4 (e+f x))+18 \sin (e+f x)-32 \sin (2 (e+f x))-30 \sin (3 (e+f x))-32 \sin (4 (e+f x))\right )}{192 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 169, normalized size = 1.74
method | result | size |
risch | \(a^{3} c^{2} x +\frac {i c^{2} a^{3} \left (15 \,{\mathrm e}^{7 i \left (f x +e \right )}-48 \,{\mathrm e}^{6 i \left (f x +e \right )}-9 \,{\mathrm e}^{5 i \left (f x +e \right )}-96 \,{\mathrm e}^{4 i \left (f x +e \right )}+9 \,{\mathrm e}^{3 i \left (f x +e \right )}-80 \,{\mathrm e}^{2 i \left (f x +e \right )}-15 \,{\mathrm e}^{i \left (f x +e \right )}-32\right )}{12 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}+\frac {3 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{8 f}-\frac {3 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{8 f}\) | \(162\) |
derivativedivides | \(\frac {c^{2} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-2 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{2} a^{3} \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{2} a^{3} \left (f x +e \right )}{f}\) | \(169\) |
default | \(\frac {c^{2} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-2 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{2} a^{3} \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{2} a^{3} \left (f x +e \right )}{f}\) | \(169\) |
norman | \(\frac {a^{3} c^{2} x +a^{3} c^{2} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 a^{3} c^{2} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+6 a^{3} c^{2} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 a^{3} c^{2} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {11 c^{2} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {137 c^{2} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}-\frac {71 c^{2} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}+\frac {5 c^{2} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {3 c^{2} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {3 c^{2} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs.
\(2 (96) = 192\).
time = 0.31, size = 219, normalized size = 2.26 \begin {gather*} \frac {16 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} + 48 \, {\left (f x + e\right )} a^{3} c^{2} - 3 \, a^{3} c^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 24 \, a^{3} c^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a^{3} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 96 \, a^{3} c^{2} \tan \left (f x + e\right )}{48 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.37, size = 157, normalized size = 1.62 \begin {gather*} \frac {48 \, a^{3} c^{2} f x \cos \left (f x + e\right )^{4} + 9 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 9 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (32 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} - 8 \, a^{3} c^{2} \cos \left (f x + e\right ) - 6 \, a^{3} c^{2}\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} c^{2} \left (\int 1\, dx + \int \sec {\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.55, size = 153, normalized size = 1.58 \begin {gather*} \frac {24 \, {\left (f x + e\right )} a^{3} c^{2} + 9 \, a^{3} c^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 9 \, a^{3} c^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 71 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 137 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 33 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.13, size = 163, normalized size = 1.68 \begin {gather*} \frac {\frac {5\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{4}-\frac {71\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{12}+\frac {137\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{12}-\frac {11\,a^3\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+a^3\,c^2\,x+\frac {3\,a^3\,c^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________