Optimal. Leaf size=318 \[ \frac {a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^5 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {5 a b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {5 a b^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14, 270} \[ \frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}+\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^5 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {5 a b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {5 a b^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 270
Rule 2606
Rule 2611
Rule 3090
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \sec ^3(c+d x)+5 a^4 b \sec ^3(c+d x) \tan (c+d x)+10 a^3 b^2 \sec ^3(c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec ^3(c+d x) \tan ^3(c+d x)+5 a b^4 \sec ^3(c+d x) \tan ^4(c+d x)+b^5 \sec ^3(c+d x) \tan ^5(c+d x)\right ) \, dx\\ &=a^5 \int \sec ^3(c+d x) \, dx+\left (5 a^4 b\right ) \int \sec ^3(c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec ^3(c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec ^3(c+d x) \tan ^5(c+d x) \, dx\\ &=\frac {a^5 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}+\frac {1}{2} a^5 \int \sec (c+d x) \, dx-\frac {1}{2} \left (5 a^3 b^2\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{2} \left (5 a b^4\right ) \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (5 a^4 b\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}+\frac {a^5 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}-\frac {1}{4} \left (5 a^3 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{8} \left (5 a b^4\right ) \int \sec ^3(c+d x) \, dx+\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec ^3(c+d x)}{3 d}+\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {a^5 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {5 a b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}+\frac {1}{16} \left (5 a b^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {5 a b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec ^3(c+d x)}{3 d}+\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {a^5 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {5 a b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 6.34, size = 1677, normalized size = 5.27 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 227, normalized size = 0.71 \[ \frac {105 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 480 \, b^{5} + 1120 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 1344 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (40 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (12 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 6.46, size = 680, normalized size = 2.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.28, size = 564, normalized size = 1.77 \[ \frac {8 b^{5} \cos \left (d x +c \right )}{105 d}-\frac {5 a \,b^{4} \sin \left (d x +c \right )}{16 d}+\frac {5 a^{3} b^{2} \sin \left (d x +c \right )}{4 d}+\frac {a^{5} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}-\frac {4 a^{2} b^{3} \cos \left (d x +c \right )}{3 d}-\frac {2 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{3 d}-\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{105 d \cos \left (d x +c \right )^{3}}-\frac {5 a^{3} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {5 a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}+\frac {2 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}+\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{24 d \cos \left (d x +c \right )^{4}}+\frac {a^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {5 a \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}-\frac {5 a \,b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{48 d}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )}+\frac {b^{5} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}+\frac {4 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) b^{5}}{105 d}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {5 a^{4} b}{3 d \cos \left (d x +c \right )^{3}}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}-\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{48 d \cos \left (d x +c \right )^{2}}+\frac {5 a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}+\frac {2 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{5}}+\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{6}}-\frac {2 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 289, normalized size = 0.91 \[ -\frac {175 \, a b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} + 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2100 \, a^{3} b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 840 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {5600 \, a^{4} b}{\cos \left (d x + c\right )^{3}} + \frac {2240 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{5}} - \frac {32 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} b^{5}}{\cos \left (d x + c\right )^{7}}}{3360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.21, size = 514, normalized size = 1.62 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^5-\frac {5\,a^3\,b^2}{2}+\frac {5\,a\,b^4}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-4\,a^5+10\,a^3\,b^2+\frac {25\,a\,b^4}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (a^5+\frac {5\,a^3\,b^2}{2}-\frac {5\,a\,b^4}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-4\,a^5+10\,a^3\,b^2+\frac {25\,a\,b^4}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (5\,a^5-\frac {55\,a^3\,b^2}{2}+\frac {485\,a\,b^4}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (5\,a^5-\frac {55\,a^3\,b^2}{2}+\frac {485\,a\,b^4}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (30\,a^4\,b-16\,a^2\,b^3+\frac {16\,b^5}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {40\,a^4\,b}{3}-\frac {56\,a^2\,b^3}{3}+\frac {16\,b^5}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-\frac {160\,a^4\,b}{3}+\frac {80\,a^2\,b^3}{3}+\frac {16\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {190\,a^4\,b}{3}-\frac {200\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+\frac {10\,a^4\,b}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^5+\frac {5\,a^3\,b^2}{2}-\frac {5\,a\,b^4}{8}\right )+\frac {16\,b^5}{105}-\frac {8\,a^2\,b^3}{3}+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________