Optimal. Leaf size=91 \[ -\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2875, 2873, 2607, 14, 2606, 270} \[ -\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\int (-a+a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^4(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^5(c+d x)+a^2 \cot ^2(c+d x) \csc ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^4(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^3(c+d x) \csc ^5(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \operatorname {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \operatorname {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.70, size = 149, normalized size = 1.64 \[ -\frac {\csc (c) (-714 \sin (c+d x)-408 \sin (2 (c+d x))+153 \sin (3 (c+d x))+204 \sin (4 (c+d x))+51 \sin (5 (c+d x))+1680 \sin (2 c+d x)+128 \sin (c+2 d x)-48 \sin (2 c+3 d x)-64 \sin (3 c+4 d x)-16 \sin (4 c+5 d x)+1344 \sin (c)-1456 \sin (d x)) \csc ^3(c+d x) \sec ^2(c+d x)}{13440 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 108, normalized size = 1.19 \[ \frac {2 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )^{2} + 24 \, \cos \left (d x + c\right ) + 12}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 105, normalized size = 1.15 \[ -\frac {\frac {35 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {15 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{14}}}{3360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.80, size = 86, normalized size = 0.95 \[ \frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 134, normalized size = 1.47 \[ -\frac {\frac {\frac {210 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2}} + \frac {35 \, {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{2} \sin \left (d x + c\right )^{3}}}{3360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 121, normalized size = 1.33 \[ -\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+54\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-15}{3360\,a^2\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________