Optimal. Leaf size=52 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} d \sqrt {a+b}}-\frac {\cos (c+d x)}{b d} \]
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Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3186, 388, 208} \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} d \sqrt {a+b}}-\frac {\cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x)}{b d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{b d}\\ &=\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b} d}-\frac {\cos (c+d x)}{b d}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 125, normalized size = 2.40 \[ -\frac {\sqrt {b} \sqrt {-a-b} \cos (c+d x)+a \tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )+a \tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{b^{3/2} d \sqrt {-a-b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 165, normalized size = 3.17 \[ \left [\frac {\sqrt {a b + b^{2}} a \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )}{2 \, {\left (a b^{2} + b^{3}\right )} d}, -\frac {\sqrt {-a b - b^{2}} a \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) + {\left (a b + b^{2}\right )} \cos \left (d x + c\right )}{{\left (a b^{2} + b^{3}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 57, normalized size = 1.10 \[ -\frac {a \arctan \left (\frac {b \cos \left (d x + c\right )}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b d} - \frac {\cos \left (d x + c\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 45, normalized size = 0.87 \[ \frac {-\frac {\cos \left (d x +c \right )}{b}+\frac {a \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{b \sqrt {\left (a +b \right ) b}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 67, normalized size = 1.29 \[ -\frac {\frac {a \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} b} + \frac {2 \, \cos \left (d x + c\right )}{b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 44, normalized size = 0.85 \[ \frac {a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )}{b^{3/2}\,d\,\sqrt {a+b}}-\frac {\cos \left (c+d\,x\right )}{b\,d} \]
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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