Optimal. Leaf size=224 \[ \frac {a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 b \sec (c+d x)}{d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac {10 a^2 b^3 \sec (c+d x)}{d}+\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b^5 \sec ^5(c+d x)}{5 d}-\frac {2 b^5 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.23, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3090, 3770, 2606, 8, 2611, 194} \[ \frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac {10 a^2 b^3 \sec (c+d x)}{d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac {5 a^4 b \sec (c+d x)}{d}+\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b^5 \sec ^5(c+d x)}{5 d}-\frac {2 b^5 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 2606
Rule 2611
Rule 3090
Rule 3770
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \sec (c+d x)+5 a^4 b \sec (c+d x) \tan (c+d x)+10 a^3 b^2 \sec (c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec (c+d x) \tan ^3(c+d x)+5 a b^4 \sec (c+d x) \tan ^4(c+d x)+b^5 \sec (c+d x) \tan ^5(c+d x)\right ) \, dx\\ &=a^5 \int \sec (c+d x) \, dx+\left (5 a^4 b\right ) \int \sec (c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec (c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec (c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec (c+d x) \tan ^5(c+d x) \, dx\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{d}+\frac {5 a b^4 \sec (c+d x) \tan ^3(c+d x)}{4 d}-\left (5 a^3 b^2\right ) \int \sec (c+d x) \, dx-\frac {1}{4} \left (15 a b^4\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx+\frac {\left (5 a^4 b\right ) \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}+\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 b \sec (c+d x)}{d}-\frac {10 a^2 b^3 \sec (c+d x)}{d}+\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{d}-\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec (c+d x) \tan ^3(c+d x)}{4 d}+\frac {1}{8} \left (15 a b^4\right ) \int \sec (c+d x) \, dx+\frac {b^5 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a^5 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {5 a^4 b \sec (c+d x)}{d}-\frac {10 a^2 b^3 \sec (c+d x)}{d}+\frac {b^5 \sec (c+d x)}{d}+\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac {2 b^5 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{d}-\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec (c+d x) \tan ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 6.29, size = 1219, normalized size = 5.44 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 196, normalized size = 0.88 \[ \frac {15 \, {\left (8 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{5} + 240 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 160 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 150 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + {\left (8 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 410, normalized size = 1.83 \[ \frac {15 \, {\left (8 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (8 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (600 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 225 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 600 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1200 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2400 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2400 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3600 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5600 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 640 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1200 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1050 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2400 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4000 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 320 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 600 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 225 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a^{4} b + 800 \, a^{2} b^{3} - 64 \, b^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 440, normalized size = 1.96 \[ \frac {a^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {5 a^{4} b}{d \cos \left (d x +c \right )}+\frac {5 a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {5 a^{3} b^{2} \sin \left (d x +c \right )}{d}-\frac {5 a^{3} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {10 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {10 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )}-\frac {10 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{3 d}-\frac {20 a^{2} b^{3} \cos \left (d x +c \right )}{3 d}+\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {5 a \,b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {15 a \,b^{4} \sin \left (d x +c \right )}{8 d}+\frac {15 a \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{5 d \cos \left (d x +c \right )^{5}}-\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{15 d \cos \left (d x +c \right )^{3}}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{5 d \cos \left (d x +c \right )}+\frac {8 b^{5} \cos \left (d x +c \right )}{15 d}+\frac {b^{5} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) b^{5}}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 230, normalized size = 1.03 \[ \frac {75 \, a b^{4} {\left (\frac {2 \, {\left (5 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 600 \, a^{3} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 120 \, a^{5} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {1200 \, a^{4} b}{\cos \left (d x + c\right )} - \frac {800 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{3}} + \frac {16 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} + 3\right )} b^{5}}{\cos \left (d x + c\right )^{5}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.26, size = 345, normalized size = 1.54 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^5-10\,a^3\,b^2+\frac {15\,a\,b^4}{4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {15\,a\,b^4}{4}-10\,a^3\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {35\,a\,b^4}{2}-20\,a^3\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {35\,a\,b^4}{2}-20\,a^3\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (40\,a^4\,b-\frac {200\,a^2\,b^3}{3}+\frac {16\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (60\,a^4\,b-\frac {280\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+10\,a^4\,b+\frac {16\,b^5}{15}-\frac {40\,a^2\,b^3}{3}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,a\,b^4}{4}-10\,a^3\,b^2\right )+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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