Optimal. Leaf size=204 \[ \frac {a^5 \sin (c+d x)}{d}-\frac {5 a^4 b \cos (c+d x)}{d}-\frac {10 a^3 b^2 \sin (c+d x)}{d}+\frac {10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {10 a^2 b^3 \cos (c+d x)}{d}+\frac {10 a^2 b^3 \sec (c+d x)}{d}+\frac {15 a b^4 \sin (c+d x)}{2 d}+\frac {5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^5 \cos (c+d x)}{d}+\frac {b^5 \sec ^3(c+d x)}{3 d}-\frac {2 b^5 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.21, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3090, 2637, 2638, 2592, 321, 206, 2590, 14, 288, 270} \[ -\frac {10 a^3 b^2 \sin (c+d x)}{d}+\frac {10 a^2 b^3 \cos (c+d x)}{d}+\frac {10 a^2 b^3 \sec (c+d x)}{d}+\frac {10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {5 a^4 b \cos (c+d x)}{d}+\frac {a^5 \sin (c+d x)}{d}+\frac {15 a b^4 \sin (c+d x)}{2 d}+\frac {5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^5 \cos (c+d x)}{d}+\frac {b^5 \sec ^3(c+d x)}{3 d}-\frac {2 b^5 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 206
Rule 270
Rule 288
Rule 321
Rule 2590
Rule 2592
Rule 2637
Rule 2638
Rule 3090
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos (c+d x)+5 a^4 b \sin (c+d x)+10 a^3 b^2 \sin (c+d x) \tan (c+d x)+10 a^2 b^3 \sin (c+d x) \tan ^2(c+d x)+5 a b^4 \sin (c+d x) \tan ^3(c+d x)+b^5 \sin (c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^5 \int \cos (c+d x) \, dx+\left (5 a^4 b\right ) \int \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sin (c+d x) \tan (c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sin (c+d x) \tan ^2(c+d x) \, dx+\left (5 a b^4\right ) \int \sin (c+d x) \tan ^3(c+d x) \, dx+b^5 \int \sin (c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac {5 a^4 b \cos (c+d x)}{d}+\frac {a^5 \sin (c+d x)}{d}+\frac {\left (10 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (5 a b^4\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {5 a^4 b \cos (c+d x)}{d}+\frac {a^5 \sin (c+d x)}{d}-\frac {10 a^3 b^2 \sin (c+d x)}{d}+\frac {5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}+\frac {\left (10 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (15 a b^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}-\frac {b^5 \operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {5 a^4 b \cos (c+d x)}{d}+\frac {10 a^2 b^3 \cos (c+d x)}{d}-\frac {b^5 \cos (c+d x)}{d}+\frac {10 a^2 b^3 \sec (c+d x)}{d}-\frac {2 b^5 \sec (c+d x)}{d}+\frac {b^5 \sec ^3(c+d x)}{3 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {10 a^3 b^2 \sin (c+d x)}{d}+\frac {15 a b^4 \sin (c+d x)}{2 d}+\frac {5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {\left (15 a b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}\\ &=\frac {10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^4 b \cos (c+d x)}{d}+\frac {10 a^2 b^3 \cos (c+d x)}{d}-\frac {b^5 \cos (c+d x)}{d}+\frac {10 a^2 b^3 \sec (c+d x)}{d}-\frac {2 b^5 \sec (c+d x)}{d}+\frac {b^5 \sec ^3(c+d x)}{3 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {10 a^3 b^2 \sin (c+d x)}{d}+\frac {15 a b^4 \sin (c+d x)}{2 d}+\frac {5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 5.90, size = 397, normalized size = 1.95 \[ \frac {120 a^2 b^3-30 a b^2 \left (4 a^2-3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a b^2 \left (4 a^2-3 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 b^3 \left (60 a^2-11 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 b^3 \left (11 b^2-60 a^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+12 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (c+d x)-12 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x)+\frac {b^4 (15 a+b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^4 (b-15 a)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {2 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}-22 b^5}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 190, normalized size = 0.93 \[ \frac {4 \, b^{5} - 12 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 24 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (5 \, a b^{4} \cos \left (d x + c\right ) + 2 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 281, normalized size = 1.38 \[ \frac {15 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {12 \, {\left (a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (15 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, a^{2} b^{3} + 10 \, b^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 327, normalized size = 1.60 \[ \frac {a^{5} \sin \left (d x +c \right )}{d}-\frac {5 a^{4} b \cos \left (d x +c \right )}{d}-\frac {10 a^{3} b^{2} \sin \left (d x +c \right )}{d}+\frac {10 a^{3} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {10 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {10 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{d}+\frac {20 a^{2} b^{3} \cos \left (d x +c \right )}{d}+\frac {5 a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {5 a \,b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {15 a \,b^{4} \sin \left (d x +c \right )}{2 d}-\frac {15 a \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}-\frac {8 b^{5} \cos \left (d x +c \right )}{3 d}-\frac {b^{5} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {4 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) b^{5}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 181, normalized size = 0.89 \[ -\frac {15 \, a b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 120 \, a^{2} b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 4 \, b^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - 60 \, a^{3} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 60 \, a^{4} b \cos \left (d x + c\right ) - 12 \, a^{5} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.05, size = 302, normalized size = 1.48 \[ -\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (15\,a\,b^4-20\,a^3\,b^2\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^5-20\,a^3\,b^2+15\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (30\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (2\,a^5-20\,a^3\,b^2+15\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a^5-60\,a^3\,b^2+25\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^5-60\,a^3\,b^2+25\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (30\,a^4\,b-80\,a^2\,b^3+\frac {32\,b^5}{3}\right )-10\,a^4\,b-\frac {16\,b^5}{3}+40\,a^2\,b^3+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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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