Optimal. Leaf size=131 \[ \frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d} \]
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Rubi [A]
time = 0.13, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3302, 1185,
1181, 211, 214} \begin {gather*} \frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}+\frac {\sin ^3(c+d x)}{3 b d}-\frac {3 \sin (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 1181
Rule 1185
Rule 3302
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {3}{b}+\frac {x^2}{b}+\frac {3 a+b-(a+3 b) x^2}{b \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d}+\frac {\text {Subst}\left (\int \frac {3 a+b+(-a-3 b) x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{b d}\\ &=-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} b d}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} b d}\\ &=\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 207, normalized size = 1.58 \begin {gather*} \frac {3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )-3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )-36 a^{3/4} b^{3/4} \sin (c+d x)+4 a^{3/4} b^{3/4} \sin ^3(c+d x)}{12 a^{3/4} b^{7/4} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.16, size = 176, normalized size = 1.34
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (3 a +b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (-a -3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\) | \(176\) |
default | \(\frac {\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (3 a +b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (-a -3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\) | \(176\) |
risch | \(\frac {11 i {\mathrm e}^{i \left (d x +c \right )}}{8 b d}-\frac {11 i {\mathrm e}^{-i \left (d x +c \right )}}{8 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (256 a^{3} b^{7} d^{4} \textit {\_Z}^{4}+\left (192 a^{4} b^{4} d^{2}+640 a^{3} b^{5} d^{2}+192 a^{2} b^{6} d^{2}\right ) \textit {\_Z}^{2}-a^{6}+6 a^{5} b -15 a^{4} b^{2}+20 a^{3} b^{3}-15 a^{2} b^{4}+6 a \,b^{5}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {128 i a^{4} b^{5} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {384 i a^{3} b^{6} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {72 i a^{5} b^{2} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {672 i a^{4} b^{3} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {1008 i a^{3} b^{4} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {288 i a^{2} b^{5} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {8 i a \,b^{6} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {12 a^{5} b}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {27 a^{4} b^{2}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {27 a^{2} b^{4}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {12 a \,b^{5}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {b^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right )\right )-\frac {\sin \left (3 d x +3 c \right )}{12 b d}\) | \(806\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 177, normalized size = 1.35 \begin {gather*} \frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )\right )}}{b} + \frac {3 \, {\left (\frac {2 \, {\left (b {\left (3 \, \sqrt {a} + \sqrt {b}\right )} + a^{\frac {3}{2}} + 3 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b {\left (3 \, \sqrt {a} - \sqrt {b}\right )} + a^{\frac {3}{2}} - 3 \, a \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{b}}{12 \, d} \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. 1429 vs.
\(2 (101) = 202\).
time = 0.61, size = 1429, normalized size = 10.91 \begin {gather*} \frac {3 \, b d \sqrt {-\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left ({\left (a^{4} b^{5} + 3 \, a^{3} b^{6}\right )} d^{3} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - {\left (3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right )} d\right )} \sqrt {-\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}}\right ) - 3 \, b d \sqrt {\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left ({\left (a^{4} b^{5} + 3 \, a^{3} b^{6}\right )} d^{3} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + {\left (3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right )} d\right )} \sqrt {\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}}\right ) - 3 \, b d \sqrt {-\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left ({\left (a^{4} b^{5} + 3 \, a^{3} b^{6}\right )} d^{3} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - {\left (3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right )} d\right )} \sqrt {-\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}}\right ) + 3 \, b d \sqrt {\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left ({\left (a^{4} b^{5} + 3 \, a^{3} b^{6}\right )} d^{3} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + {\left (3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right )} d\right )} \sqrt {\frac {a b^{3} d^{2} \sqrt {\frac {a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}}\right ) - 4 \, {\left (\cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{12 \, b 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. 360 vs.
\(2 (101) = 202\).
time = 0.81, size = 360, normalized size = 2.75 \begin {gather*} \frac {\frac {8 \, {\left (b^{2} \sin \left (d x + c\right )^{3} - 9 \, b^{2} \sin \left (d x + c\right )\right )}}{b^{3}} - \frac {6 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} - \frac {6 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.77, size = 1931, normalized size = 14.74 \begin {gather*} \frac {{\sin \left (c+d\,x\right )}^3}{3\,b\,d}-\frac {3\,\sin \left (c+d\,x\right )}{b\,d}+\frac {\mathrm {atan}\left (\frac {a^3\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}-\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,8{}\mathrm {i}}{92\,a\,b+\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}+\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}+\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}+\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}+\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {b^3\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}-\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,8{}\mathrm {i}}{92\,a\,b+\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}+\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}+\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}+\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}+\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}-\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,120{}\mathrm {i}}{92\,a\,b+\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}+\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}+\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}+\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}+\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {a^2\,b\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}-\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}-\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,120{}\mathrm {i}}{92\,a\,b+\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}+\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}+\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}+\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}+\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}\right )\,\sqrt {-\frac {a^3\,\sqrt {a^3\,b^7}+b^3\,\sqrt {a^3\,b^7}+6\,a^2\,b^6+20\,a^3\,b^5+6\,a^4\,b^4+15\,a\,b^2\,\sqrt {a^3\,b^7}+15\,a^2\,b\,\sqrt {a^3\,b^7}}{16\,a^3\,b^7}}\,2{}\mathrm {i}}{d}+\frac {\mathrm {atan}\left (\frac {a^3\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}+\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,8{}\mathrm {i}}{92\,a\,b-\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}-\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}-\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}-\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}-\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {b^3\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}+\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,8{}\mathrm {i}}{92\,a\,b-\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}-\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}-\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}-\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}-\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}+\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,120{}\mathrm {i}}{92\,a\,b-\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}-\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}-\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}-\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}-\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}+\frac {a^2\,b\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^7}}{16\,b^7}-\frac {3\,a}{8\,b^3}-\frac {5}{4\,b^2}-\frac {3}{8\,a\,b}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a\,b^6}+\frac {15\,\sqrt {a^3\,b^7}}{16\,a^2\,b^5}+\frac {\sqrt {a^3\,b^7}}{16\,a^3\,b^4}}\,120{}\mathrm {i}}{92\,a\,b-\frac {120\,\sqrt {a^3\,b^7}}{b^3}+120\,a^2+6\,b^2+\frac {36\,a^3}{b}+\frac {2\,a^4}{b^2}-\frac {36\,\sqrt {a^3\,b^7}}{a\,b^2}-\frac {2\,\sqrt {a^3\,b^7}}{a^2\,b}-\frac {6\,a^2\,\sqrt {a^3\,b^7}}{b^5}-\frac {92\,a\,\sqrt {a^3\,b^7}}{b^4}}\right )\,\sqrt {\frac {a^3\,\sqrt {a^3\,b^7}+b^3\,\sqrt {a^3\,b^7}-6\,a^2\,b^6-20\,a^3\,b^5-6\,a^4\,b^4+15\,a\,b^2\,\sqrt {a^3\,b^7}+15\,a^2\,b\,\sqrt {a^3\,b^7}}{16\,a^3\,b^7}}\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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