Optimal. Leaf size=82 \[ -\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {15 \sec (c+d x)}{8 a d}-\frac {5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3254, 2702,
294, 327, 213} \begin {gather*} \frac {15 \sec (c+d x)}{8 a d}-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d}-\frac {5 \csc ^2(c+d x) \sec (c+d x)}{8 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 213
Rule 294
Rule 327
Rule 2702
Rule 3254
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \csc ^5(c+d x) \sec ^2(c+d x) \, dx}{a}\\ &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^3} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{4 a d}\\ &=-\frac {5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d}+\frac {15 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{8 a d}\\ &=\frac {15 \sec (c+d x)}{8 a d}-\frac {5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d}+\frac {15 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{8 a d}\\ &=-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {15 \sec (c+d x)}{8 a d}-\frac {5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}-\frac {\csc ^4(c+d x) \sec (c+d x)}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.78, size = 132, normalized size = 1.61 \begin {gather*} -\frac {14 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (78+\cos (c+d x) \left (-8 \left (8+15 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )\right )-14 \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}{64 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 87, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {1}{\cos \left (d x +c \right )}+\frac {1}{16 \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {7}{16 \left (1+\cos \left (d x +c \right )\right )}-\frac {15 \ln \left (1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{16 \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {7}{16 \left (\cos \left (d x +c \right )-1\right )}+\frac {15 \ln \left (\cos \left (d x +c \right )-1\right )}{16}}{d a}\) | \(87\) |
default | \(\frac {\frac {1}{\cos \left (d x +c \right )}+\frac {1}{16 \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {7}{16 \left (1+\cos \left (d x +c \right )\right )}-\frac {15 \ln \left (1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{16 \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {7}{16 \left (\cos \left (d x +c \right )-1\right )}+\frac {15 \ln \left (\cos \left (d x +c \right )-1\right )}{16}}{d a}\) | \(87\) |
norman | \(\frac {\frac {1}{64 a d}+\frac {15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}+\frac {15 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}\) | \(132\) |
risch | \(\frac {15 \,{\mathrm e}^{9 i \left (d x +c \right )}-40 \,{\mathrm e}^{7 i \left (d x +c \right )}+18 \,{\mathrm e}^{5 i \left (d x +c \right )}-40 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}}{4 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 90, normalized size = 1.10 \begin {gather*} \frac {\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 25 \, \cos \left (d x + c\right )^{2} + 8\right )}}{a \cos \left (d x + c\right )^{5} - 2 \, a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )} - \frac {15 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 135, normalized size = 1.65 \begin {gather*} \frac {30 \, \cos \left (d x + c\right )^{4} - 50 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 16}{16 \, {\left (a d \cos \left (d x + c\right )^{5} - 2 \, a d \cos \left (d x + c\right )^{3} + a 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 {\csc ^{5}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs.
\(2 (74) = 148\).
time = 0.44, size = 181, normalized size = 2.21 \begin {gather*} \frac {\frac {{\left (\frac {16 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {90 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {60 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} - \frac {\frac {16 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2}} + \frac {128}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.10, size = 74, normalized size = 0.90 \begin {gather*} \frac {\frac {15\,{\cos \left (c+d\,x\right )}^4}{8}-\frac {25\,{\cos \left (c+d\,x\right )}^2}{8}+1}{d\,\left (a\,{\cos \left (c+d\,x\right )}^5-2\,a\,{\cos \left (c+d\,x\right )}^3+a\,\cos \left (c+d\,x\right )\right )}-\frac {15\,\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{8\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________