Optimal. Leaf size=125 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]
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Rubi [A] time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3215, 1093, 205, 208} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 1093
Rule 3215
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{a-b \sin ^4(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 \sqrt {a} d}-\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 \sqrt {a} d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt [4]{b} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt [4]{b} d}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 183, normalized size = 1.46 \[ \frac {i \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \text {$\#$1} \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 \text {$\#$1} \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^6 b-3 \text {$\#$1}^4 b-8 \text {$\#$1}^2 a+3 \text {$\#$1}^2 b-b}\& \right ]}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 703, normalized size = 5.62 \[ -\frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - a d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} + \cos \left (d x + c\right )\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - a d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} - \cos \left (d x + c\right )\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + a d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} + \cos \left (d x + c\right )\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + a d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} - \cos \left (d x + c\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.76, size = 183, normalized size = 1.46 \[ -\frac {\sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {b d^{2} + \sqrt {{\left (a - b\right )} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right )}{2 \, {\left (a b + \sqrt {a b} a\right )} d {\left | b \right |}} + \frac {\sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {b d^{2} - \sqrt {{\left (a - b\right )} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right )}{2 \, {\left (a b - \sqrt {a b} a\right )} d {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 90, normalized size = 0.72 \[ -\frac {b \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {b \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}} \]
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 \[ -\int \frac {\sin \left (d x + c\right )}{b \sin \left (d x + c\right )^{4} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.11, size = 361, normalized size = 2.89 \[ \frac {\ln \left (4\,a\,b^3\,\sqrt {\frac {1}{a\,b+\sqrt {a^3\,b}}}-4\,b^3\,\cos \left (c+d\,x\right )+\frac {4\,a\,b^4\,\cos \left (c+d\,x\right )}{a\,b+\sqrt {a^3\,b}}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{16\,\left (a^3\,b-a^2\,b^2\right )}}}{d}+\frac {\ln \left (4\,b^3\,\cos \left (c+d\,x\right )-4\,a\,b^3\,\sqrt {\frac {1}{a\,b-\sqrt {a^3\,b}}}-\frac {4\,a\,b^4\,\cos \left (c+d\,x\right )}{a\,b-\sqrt {a^3\,b}}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{16\,\left (a^3\,b-a^2\,b^2\right )}}}{d}-\frac {\ln \left (4\,b^3\,\cos \left (c+d\,x\right )+4\,a\,b^3\,\sqrt {\frac {1}{a\,b+\sqrt {a^3\,b}}}-\frac {4\,a\,b^4\,\cos \left (c+d\,x\right )}{a\,b+\sqrt {a^3\,b}}\right )\,\sqrt {\frac {1}{a\,b+\sqrt {a^3\,b}}}}{4\,d}-\frac {\ln \left (4\,b^3\,\cos \left (c+d\,x\right )+4\,a\,b^3\,\sqrt {\frac {1}{a\,b-\sqrt {a^3\,b}}}-\frac {4\,a\,b^4\,\cos \left (c+d\,x\right )}{a\,b-\sqrt {a^3\,b}}\right )\,\sqrt {\frac {1}{a\,b-\sqrt {a^3\,b}}}}{4\,d} \]
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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