Optimal. Leaf size=184 \[ \frac {5 x}{8 b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d} \]
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Rubi [A]
time = 0.20, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 1301,
205, 209, 1180, 211} \begin {gather*} \frac {a^{5/4} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{5/4} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {x (a+b)}{b^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 b d}+\frac {5 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac {5 x}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 209
Rule 211
Rule 1180
Rule 1301
Rule 3296
Rubi steps
\begin {align*} \int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^3 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )^3}+\frac {2}{b \left (1+x^2\right )^2}+\frac {-a-b}{b^2 \left (1+x^2\right )}+\frac {a^2 \left (1+x^2\right )}{b^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a^2 \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=-\frac {(a+b) x}{b^2}+\frac {\cos (c+d x) \sin (c+d x)}{b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\left (a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}+\frac {\left (a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}-\frac {3 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 b d}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=\frac {x}{b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 b d}\\ &=\frac {5 x}{8 b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 172, normalized size = 0.93 \begin {gather*} -\frac {4 (8 a+3 b) (c+d x)-\frac {16 a^{3/2} \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {16 a^{3/2} \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-8 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 209, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}-\frac {\frac {-\frac {5 b \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {3 b \tan \left (d x +c \right )}{8}}{\left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {\left (8 a +3 b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}}{d}\) | \(209\) |
default | \(\frac {\frac {a^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}-\frac {\frac {-\frac {5 b \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {3 b \tan \left (d x +c \right )}{8}}{\left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {\left (8 a +3 b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}}{d}\) | \(209\) |
risch | \(-\frac {a x}{b^{2}}-\frac {3 x}{8 b}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a \,b^{8} d^{4}-b^{9} d^{4}\right ) \textit {\_Z}^{4}+8192 a^{3} b^{4} d^{2} \textit {\_Z}^{2}+16777216 a^{5}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {i b^{5} d^{3}}{131072 a^{2}}-\frac {i b^{6} d^{3}}{131072 a^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {b^{3} d^{2}}{2048 a}+\frac {b^{4} d^{2}}{2048 a^{2}}\right ) \textit {\_R}^{2}+\left (\frac {i b d}{32}+\frac {i b^{2} d}{32 a}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )}{256}-\frac {\sin \left (4 d x +4 c \right )}{32 b d}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1311 vs.
\(2 (142) = 284\).
time = 0.56, size = 1311, normalized size = 7.12 \begin {gather*} -\frac {b^{2} d \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (\frac {1}{4} \, a^{3} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, a^{3} - \frac {1}{4} \, {\left (2 \, {\left (a^{2} b^{3} - a b^{4}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{2} b^{3} - a b^{4}\right )} d^{2}\right )} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{2} b^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) - b^{2} d \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (\frac {1}{4} \, a^{3} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, a^{3} - \frac {1}{4} \, {\left (2 \, {\left (a^{2} b^{3} - a b^{4}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{2} b^{3} - a b^{4}\right )} d^{2}\right )} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{2} b^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) - b^{2} d \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (-\frac {1}{4} \, a^{3} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, a^{3} - \frac {1}{4} \, {\left (2 \, {\left (a^{2} b^{3} - a b^{4}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{2} b^{3} - a b^{4}\right )} d^{2}\right )} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{2} b^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) + b^{2} d \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (-\frac {1}{4} \, a^{3} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, a^{3} - \frac {1}{4} \, {\left (2 \, {\left (a^{2} b^{3} - a b^{4}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{2} b^{3} - a b^{4}\right )} d^{2}\right )} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{2} b^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) + {\left (8 \, a + 3 \, b\right )} d x + {\left (2 \, b \cos \left (d x + c\right )^{3} - 5 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (142) = 284\).
time = 1.11, size = 461, normalized size = 2.51 \begin {gather*} \frac {\frac {4 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} + \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} + \frac {4 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} - \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} - \frac {{\left (d x + c\right )} {\left (8 \, a + 3 \, b\right )}}{b^{2}} + \frac {5 \, \tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} b}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.87, size = 2500, normalized size = 13.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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