Optimal. Leaf size=320 \[ -\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}-\frac {a b^7 \left (13-\frac {3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]
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Rubi [A] time = 0.30, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2668, 739, 819, 801, 633, 31} \[ -\frac {a b^7 \left (13-\frac {3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac {5 b^4 \left (-42 a^2 b^2+9 a^4-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^3 \left (-77 a^2 b^2+15 a^4-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^2 \left (-35 a^4 b^2-84 a^2 b^4+6 a^6-7 b^6\right ) \sin (c+d x)}{8 d}-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 739
Rule 801
Rule 819
Rule 2668
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {(a+x)^8}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^6 \left (-3 a^2+7 b^2+4 a x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^4 \left (3 a^4+2 a^2 b^2+35 b^4-4 a \left (3 a^2-13 b^2\right ) x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \operatorname {Subst}\left (\int \left (5 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right )+4 a \left (15 a^4-77 a^2 b^2-48 b^4\right ) x+5 \left (9 a^4-42 a^2 b^2-7 b^4\right ) x^2+4 a \left (3 a^2-13 b^2\right ) x^3+\frac {3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {b \operatorname {Subst}\left (\int \frac {3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac {\left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}-\frac {\left ((a-b)^6 \left (3 a^2+18 a b+35 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}\\ \end {align*}
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Mathematica [A] time = 4.12, size = 514, normalized size = 1.61 \[ -\frac {3 \left (a^2-b^2\right )^2 \left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))-(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)\right )+12 \left (a^2-b^2\right ) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9-6 a b^9 \left (3 a^2+11 b^2\right ) \sin ^8(c+d x)+6 \sec ^2(c+d x) (a+b \sin (c+d x))^9 \left (-a \left (3 a^2+11 b^2\right ) \sin (c+d x)+9 a^2 b+5 b^3\right )+6 b^8 \left (-27 a^4-90 a^2 b^2+5 b^4\right ) \sin ^7(c+d x)-24 a b^7 \left (27 a^4+79 a^2 b^2-8 b^4\right ) \sin ^6(c+d x)+42 b^6 \left (-36 a^6-87 a^4 b^2+10 a^2 b^4+b^6\right ) \sin ^5(c+d x)-12 a b^5 \left (189 a^6+333 a^4 b^2-8 a^2 b^4-24 b^6\right ) \sin ^4(c+d x)+14 b^4 \left (-162 a^8-144 a^6 b^2-85 a^4 b^4+50 a^2 b^6+5 b^8\right ) \sin ^3(c+d x)-24 a b^3 \left (63 a^8-21 a^6 b^2+88 a^4 b^4-8 a^2 b^6-24 b^8\right ) \sin ^2(c+d x)+6 b^2 \left (-108 a^{10}+234 a^8 b^2-28 a^6 b^4-595 a^4 b^6+350 a^2 b^8+35 b^{10}\right ) \sin (c+d x)}{48 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 366, normalized size = 1.14 \[ \frac {192 \, a b^{7} \cos \left (d x + c\right )^{6} - 96 \, a b^{7} \cos \left (d x + c\right )^{4} + 96 \, a^{7} b + 672 \, a^{5} b^{3} + 672 \, a^{3} b^{5} + 96 \, a b^{7} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 6 \, a^{8} + 168 \, a^{6} b^{2} + 420 \, a^{4} b^{4} + 168 \, a^{2} b^{6} + 6 \, b^{8} - 16 \, {\left (42 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 429, normalized size = 1.34 \[ -\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 1344 \, a^{2} b^{6} \sin \left (d x + c\right ) + 144 \, b^{8} \sin \left (d x + c\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (336 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 144 \, a b^{7} \sin \left (d x + c\right )^{4} - 3 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} + 350 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 252 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 13 \, b^{8} \sin \left (d x + c\right )^{3} + 224 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} - 224 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} - 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) - 210 \, a^{4} b^{4} \sin \left (d x + c\right ) - 196 \, a^{2} b^{6} \sin \left (d x + c\right ) - 11 \, b^{8} \sin \left (d x + c\right ) + 16 \, a^{7} b - 112 \, a^{5} b^{3} + 64 \, a b^{7}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 760, normalized size = 2.38 \[ -\frac {5 b^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{8 d}-\frac {21 a^{2} b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d}-\frac {35 a^{2} b^{6} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}-\frac {6 a \,b^{7} \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {12 a \,b^{7} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {35 a^{4} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {35 \sin \left (d x +c \right ) b^{8}}{8 d}+\frac {35 b^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {3 a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {7 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}-\frac {35 a^{4} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}-\frac {21 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {14 a^{5} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}+\frac {7 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}+\frac {35 a^{4} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}+\frac {7 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}+\frac {2 a \,b^{7} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}-\frac {105 a^{2} b^{6} \sin \left (d x +c \right )}{2 d}+\frac {105 a^{2} b^{6} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {24 a \,b^{7} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {56 a^{3} b^{5} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {105 a^{4} b^{4} \sin \left (d x +c \right )}{4 d}+\frac {105 a^{4} b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {7 a^{6} b^{2} \sin \left (d x +c \right )}{2 d}-\frac {7 a^{6} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {4 a \,b^{7} \left (\sin ^{6}\left (d x +c \right )\right )}{d}-\frac {35 b^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{24 d}-\frac {7 b^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {4 a \,b^{7} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}-\frac {28 a^{3} b^{5} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {a^{8} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {b^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {14 a^{3} b^{5} \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {2 a^{7} b}{d \cos \left (d x +c \right )^{4}}-\frac {5 b^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{8} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 348, normalized size = 1.09 \[ -\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 48 \, {\left (28 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac {6 \, {\left (16 \, a^{7} b - 112 \, a^{5} b^{3} - 336 \, a^{3} b^{5} - 80 \, a b^{7} - {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \sin \left (d x + c\right )^{3} + 32 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + {\left (5 \, a^{8} + 28 \, a^{6} b^{2} - 210 \, a^{4} b^{4} - 196 \, a^{2} b^{6} - 11 \, b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.48, size = 305, normalized size = 0.95 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^6\,\left (3\,a^2+18\,a\,b+35\,b^2\right )}{16\,d}-\frac {b^8\,{\sin \left (c+d\,x\right )}^3}{3\,d}-\frac {\sin \left (c+d\,x\right )\,\left (28\,a^2\,b^6+3\,b^8\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (-\frac {5\,a^8}{8}-\frac {7\,a^6\,b^2}{2}+\frac {105\,a^4\,b^4}{4}+\frac {49\,a^2\,b^6}{2}+\frac {11\,b^8}{8}\right )-{\sin \left (c+d\,x\right )}^3\,\left (-\frac {3\,a^8}{8}+\frac {7\,a^6\,b^2}{2}+\frac {175\,a^4\,b^4}{4}+\frac {63\,a^2\,b^6}{2}+\frac {13\,b^8}{8}\right )+10\,a\,b^7-2\,a^7\,b-{\sin \left (c+d\,x\right )}^2\,\left (28\,a^5\,b^3+56\,a^3\,b^5+12\,a\,b^7\right )+42\,a^3\,b^5+14\,a^5\,b^3}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}-\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^2}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^6\,\left (3\,a^2-18\,a\,b+35\,b^2\right )}{16\,d} \]
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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