Optimal. Leaf size=159 \[ -\frac {4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {244 \tan (c+d x)}{105 a^4 d}-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {4 \tan (c+d x)}{a^4 d (1+\sec (c+d x))}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3901, 4104,
4093, 3872, 3855, 3852, 8} \begin {gather*} \frac {244 \tan (c+d x)}{105 a^4 d}-\frac {4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 3901
Rule 4093
Rule 4104
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {\sec ^4(c+d x) (4 a-8 a \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^3(c+d x) \left (36 a^2-52 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^2(c+d x) \left (176 a^3-244 a^3 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \sec (c+d x) \left (-420 a^4+244 a^4 \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {244 \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac {4 \int \sec (c+d x) \, dx}{a^4}\\ &=-\frac {4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {244 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac {4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {244 \tan (c+d x)}{105 a^4 d}-\frac {88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(349\) vs. \(2(159)=318\).
time = 1.26, size = 349, normalized size = 2.19 \begin {gather*} \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (107520 \cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac {c}{2}\right ) \sec (c) \sec (c+d x) \left (-10780 \sin \left (\frac {d x}{2}\right )+18788 \sin \left (\frac {3 d x}{2}\right )-20524 \sin \left (c-\frac {d x}{2}\right )+14644 \sin \left (c+\frac {d x}{2}\right )-16660 \sin \left (2 c+\frac {d x}{2}\right )-4690 \sin \left (c+\frac {3 d x}{2}\right )+14378 \sin \left (2 c+\frac {3 d x}{2}\right )-9100 \sin \left (3 c+\frac {3 d x}{2}\right )+11668 \sin \left (c+\frac {5 d x}{2}\right )-630 \sin \left (2 c+\frac {5 d x}{2}\right )+9358 \sin \left (3 c+\frac {5 d x}{2}\right )-2940 \sin \left (4 c+\frac {5 d x}{2}\right )+4228 \sin \left (2 c+\frac {7 d x}{2}\right )+315 \sin \left (3 c+\frac {7 d x}{2}\right )+3493 \sin \left (4 c+\frac {7 d x}{2}\right )-420 \sin \left (5 c+\frac {7 d x}{2}\right )+664 \sin \left (3 c+\frac {9 d x}{2}\right )+105 \sin \left (4 c+\frac {9 d x}{2}\right )+559 \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{1680 a^4 d (1+\sec (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 118, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \,a^{4}}\) | \(118\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \,a^{4}}\) | \(118\) |
risch | \(\frac {8 i \left (105 \,{\mathrm e}^{8 i \left (d x +c \right )}+735 \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 \,{\mathrm e}^{6 i \left (d x +c \right )}+4165 \,{\mathrm e}^{5 i \left (d x +c \right )}+5131 \,{\mathrm e}^{4 i \left (d x +c \right )}+4697 \,{\mathrm e}^{3 i \left (d x +c \right )}+2917 \,{\mathrm e}^{2 i \left (d x +c \right )}+1057 \,{\mathrm e}^{i \left (d x +c \right )}+166\right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{4} d}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 186, normalized size = 1.17 \begin {gather*} \frac {\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.29, size = 234, normalized size = 1.47 \begin {gather*} -\frac {210 \, {\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 210 \, {\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (664 \, \cos \left (d x + c\right )^{4} + 2236 \, \cos \left (d x + c\right )^{3} + 2636 \, \cos \left (d x + c\right )^{2} + 1184 \, \cos \left (d x + c\right ) + 105\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \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*} \frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.47, size = 139, normalized size = 0.87 \begin {gather*} -\frac {\frac {3360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.70, size = 130, normalized size = 0.82 \begin {gather*} \frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4\,d}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^4\,d}-\frac {8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {49\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________