Optimal. Leaf size=97 \[ \frac {\sec ^5(c+d x)}{5 a d}-\frac {\sec ^4(c+d x)}{4 a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^2(c+d x)}{a d}+\frac {\sec (c+d x)}{a d}+\frac {\log (\cos (c+d x))}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {\sec ^5(c+d x)}{5 a d}-\frac {\sec ^4(c+d x)}{4 a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^2(c+d x)}{a d}+\frac {\sec (c+d x)}{a d}+\frac {\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^7(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^3 (a+a x)^2}{x^6} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}-\frac {a^5}{x^5}-\frac {2 a^5}{x^4}+\frac {2 a^5}{x^3}+\frac {a^5}{x^2}-\frac {a^5}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac {\log (\cos (c+d x))}{a d}+\frac {\sec (c+d x)}{a d}+\frac {\sec ^2(c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}-\frac {\sec ^4(c+d x)}{4 a d}+\frac {\sec ^5(c+d x)}{5 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 103, normalized size = 1.06 \[ \frac {\sec ^5(c+d x) (40 \cos (2 (c+d x))+60 \cos (3 (c+d x))+30 \cos (4 (c+d x))+75 \cos (3 (c+d x)) \log (\cos (c+d x))+15 \cos (5 (c+d x)) \log (\cos (c+d x))+30 \cos (c+d x) (5 \log (\cos (c+d x))+4)+58)}{240 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 75, normalized size = 0.77 \[ \frac {60 \, \cos \left (d x + c\right )^{5} \log \left (-\cos \left (d x + c\right )\right ) + 60 \, \cos \left (d x + c\right )^{4} + 60 \, \cos \left (d x + c\right )^{3} - 40 \, \cos \left (d x + c\right )^{2} - 15 \, \cos \left (d x + c\right ) + 12}{60 \, a d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 6.79, size = 201, normalized size = 2.07 \[ -\frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac {\frac {485 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1330 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1970 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {805 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 73}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.56, size = 93, normalized size = 0.96 \[ \frac {\sec ^{5}\left (d x +c \right )}{5 d a}-\frac {\sec ^{4}\left (d x +c \right )}{4 d a}-\frac {2 \left (\sec ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {\sec ^{2}\left (d x +c \right )}{d a}+\frac {\sec \left (d x +c \right )}{d a}-\frac {\ln \left (\sec \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 70, normalized size = 0.72 \[ \frac {\frac {60 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {60 \, \cos \left (d x + c\right )^{4} + 60 \, \cos \left (d x + c\right )^{3} - 40 \, \cos \left (d x + c\right )^{2} - 15 \, \cos \left (d x + c\right ) + 12}{a \cos \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.02, size = 153, normalized size = 1.58 \[ \frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {16}{15}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{7}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________