Optimal. Leaf size=102 \[ -\frac {a^2 \cos (c+d x)}{2 b^2 d (a+b) \left (a-b \cos ^2(c+d x)+b\right )}+\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} d (a+b)^{3/2}}-\frac {\cos (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.15, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3186, 390, 385, 208} \[ -\frac {a^2 \cos (c+d x)}{2 b^2 d (a+b) \left (a-b \cos ^2(c+d x)+b\right )}+\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} d (a+b)^{3/2}}-\frac {\cos (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-b x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^2}-\frac {a (a+2 b)-2 a b x^2}{b^2 \left (a+b-b x^2\right )^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x)}{b^2 d}+\frac {\operatorname {Subst}\left (\int \frac {a (a+2 b)-2 a b x^2}{\left (a+b-b x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{b^2 d}\\ &=-\frac {\cos (c+d x)}{b^2 d}-\frac {a^2 \cos (c+d x)}{2 b^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )}+\frac {(a (3 a+4 b)) \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 b^2 (a+b) d}\\ &=\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2} d}-\frac {\cos (c+d x)}{b^2 d}-\frac {a^2 \cos (c+d x)}{2 b^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.92, size = 172, normalized size = 1.69 \[ \frac {2 \sqrt {b} \cos (c+d x) \left (-\frac {a^2}{(a+b) (2 a-b \cos (2 (c+d x))+b)}-1\right )+\frac {a (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{(-a-b)^{3/2}}+\frac {a (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{(-a-b)^{3/2}}}{2 b^{5/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 427, normalized size = 4.19 \[ \left [-\frac {4 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a b + b^{2}} \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \, {\left (3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right )} \cos \left (d x + c\right )}{4 \, {\left ({\left (a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right )} d\right )}}, -\frac {2 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a b - b^{2}} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) - {\left (3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 342, normalized size = 3.35 \[ -\frac {\frac {{\left (3 \, a^{2} + 4 \, a b\right )} \arctan \left (\frac {b \cos \left (d x + c\right ) + a + b}{\sqrt {-a b - b^{2}} \cos \left (d x + c\right ) + \sqrt {-a b - b^{2}}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{2} + 2 \, a b - \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {14 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (a b^{2} + b^{3}\right )} {\left (a - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 94, normalized size = 0.92 \[ \frac {-\frac {\cos \left (d x +c \right )}{b^{2}}-\frac {a \left (-\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) \left (b \left (\cos ^{2}\left (d x +c \right )\right )-a -b \right )}-\frac {\left (3 a +4 b \right ) \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 131, normalized size = 1.28 \[ -\frac {\frac {2 \, a^{2} \cos \left (d x + c\right )}{a^{2} b^{2} + 2 \, a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{2}} + \frac {{\left (3 \, a + 4 \, b\right )} a \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b}} + \frac {4 \, \cos \left (d x + c\right )}{b^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 95, normalized size = 0.93 \[ \frac {a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )\,\left (3\,a+4\,b\right )}{2\,b^{5/2}\,d\,{\left (a+b\right )}^{3/2}}-\frac {a^2\,\cos \left (c+d\,x\right )}{2\,d\,\left (a+b\right )\,\left (-b^3\,{\cos \left (c+d\,x\right )}^2+b^3+a\,b^2\right )}-\frac {\cos \left (c+d\,x\right )}{b^2\,d} \]
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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