Optimal. Leaf size=163 \[ \frac {a^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{b^4 d \sqrt {a+b}}-\frac {\left (8 a^2-6 a b+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 b^3 d}-\frac {x \left (16 a^3-8 a^2 b+6 a b^2-5 b^3\right )}{16 b^4}+\frac {(6 a-5 b) \sin ^3(c+d x) \cos (c+d x)}{24 b^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]
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Rubi [A] time = 0.36, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3187, 470, 578, 522, 203, 205} \[ \frac {a^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{b^4 d \sqrt {a+b}}-\frac {\left (8 a^2-6 a b+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 b^3 d}-\frac {x \left (-8 a^2 b+16 a^3+6 a b^2-5 b^3\right )}{16 b^4}+\frac {(6 a-5 b) \sin ^3(c+d x) \cos (c+d x)}{24 b^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 470
Rule 522
Rule 578
Rule 3187
Rubi steps
\begin {align*} \int \frac {\sin ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}+\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (5 a+(-a+5 b) x^2\right )}{\left (1+x^2\right )^3 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{6 b d}\\ &=\frac {(6 a-5 b) \cos (c+d x) \sin ^3(c+d x)}{24 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a (6 a-5 b)-3 \left (2 a^2-a b+5 b^2\right ) x^2\right )}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{24 b^2 d}\\ &=-\frac {\left (8 a^2-6 a b+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 b^3 d}+\frac {(6 a-5 b) \cos (c+d x) \sin ^3(c+d x)}{24 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}+\frac {\operatorname {Subst}\left (\int \frac {3 a \left (8 a^2-6 a b+5 b^2\right )-3 \left (8 a^3-2 a^2 b+a b^2-5 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{48 b^3 d}\\ &=-\frac {\left (8 a^2-6 a b+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 b^3 d}+\frac {(6 a-5 b) \cos (c+d x) \sin ^3(c+d x)}{24 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}+\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{b^4 d}-\frac {\left (16 a^3-8 a^2 b+6 a b^2-5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{16 b^4 d}\\ &=-\frac {\left (16 a^3-8 a^2 b+6 a b^2-5 b^3\right ) x}{16 b^4}+\frac {a^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{b^4 \sqrt {a+b} d}-\frac {\left (8 a^2-6 a b+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 b^3 d}+\frac {(6 a-5 b) \cos (c+d x) \sin ^3(c+d x)}{24 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}\\ \end {align*}
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Mathematica [A] time = 1.18, size = 133, normalized size = 0.82 \[ -\frac {-\frac {192 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b}}+3 b \left (16 a^2-16 a b+15 b^2\right ) \sin (2 (c+d x))+12 \left (16 a^3-8 a^2 b+6 a b^2-5 b^3\right ) (c+d x)+3 b^2 (2 a-3 b) \sin (4 (c+d x))+b^3 \sin (6 (c+d x))}{192 b^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 453, normalized size = 2.78 \[ \left [\frac {12 \, a^{3} \sqrt {-\frac {a}{a + b}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 3 \, {\left (16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right )} d x - {\left (8 \, b^{3} \cos \left (d x + c\right )^{5} + 2 \, {\left (6 \, a b^{2} - 13 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{2} b - 10 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, b^{4} d}, -\frac {24 \, a^{3} \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a}{a + b}}}{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) + 3 \, {\left (16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right )} d x + {\left (8 \, b^{3} \cos \left (d x + c\right )^{5} + 2 \, {\left (6 \, a b^{2} - 13 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{2} b - 10 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, b^{4} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 233, normalized size = 1.43 \[ \frac {\frac {48 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} a^{4}}{\sqrt {a^{2} + a b} b^{4}} - \frac {3 \, {\left (16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right )} {\left (d x + c\right )}}{b^{4}} - \frac {24 \, a^{2} \tan \left (d x + c\right )^{5} - 30 \, a b \tan \left (d x + c\right )^{5} + 33 \, b^{2} \tan \left (d x + c\right )^{5} + 48 \, a^{2} \tan \left (d x + c\right )^{3} - 48 \, a b \tan \left (d x + c\right )^{3} + 40 \, b^{2} \tan \left (d x + c\right )^{3} + 24 \, a^{2} \tan \left (d x + c\right ) - 18 \, a b \tan \left (d x + c\right ) + 15 \, b^{2} \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3} b^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 361, normalized size = 2.21 \[ \frac {a^{4} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{d \,b^{4} \sqrt {a \left (a +b \right )}}-\frac {\left (\tan ^{5}\left (d x +c \right )\right ) a^{2}}{2 d \,b^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}+\frac {5 \left (\tan ^{5}\left (d x +c \right )\right ) a}{8 d \,b^{2} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {11 \left (\tan ^{5}\left (d x +c \right )\right )}{16 d b \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {\left (\tan ^{3}\left (d x +c \right )\right ) a^{2}}{d \,b^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}+\frac {\left (\tan ^{3}\left (d x +c \right )\right ) a}{d \,b^{2} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {5 \left (\tan ^{3}\left (d x +c \right )\right )}{6 d b \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {\tan \left (d x +c \right ) a^{2}}{2 d \,b^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}+\frac {3 \tan \left (d x +c \right ) a}{8 d \,b^{2} \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {5 \tan \left (d x +c \right )}{16 d b \left (\tan ^{2}\left (d x +c \right )+1\right )^{3}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \,b^{4}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{2 d \,b^{3}}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right ) a}{8 d \,b^{2}}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{16 d b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 192, normalized size = 1.18 \[ \frac {\frac {48 \, a^{4} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} b^{4}} - \frac {3 \, {\left (8 \, a^{2} - 10 \, a b + 11 \, b^{2}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (6 \, a^{2} - 6 \, a b + 5 \, b^{2}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{2} - 6 \, a b + 5 \, b^{2}\right )} \tan \left (d x + c\right )}{b^{3} \tan \left (d x + c\right )^{6} + 3 \, b^{3} \tan \left (d x + c\right )^{4} + 3 \, b^{3} \tan \left (d x + c\right )^{2} + b^{3}} - \frac {3 \, {\left (16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right )} {\left (d x + c\right )}}{b^{4}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.32, size = 2244, normalized size = 13.77 \[ \text {result too large to display} \]
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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