Optimal. Leaf size=106 \[ \frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} d \sqrt {a+b}}-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d} \]
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Rubi [A] time = 0.11, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3186, 390, 208} \[ -\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}+\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} d \sqrt {a+b}}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^2-a b+b^2}{b^3}+\frac {(a-2 b) x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b-b x^2\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{b^3 d}\\ &=\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b} d}-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d}\\ \end {align*}
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Mathematica [C] time = 1.44, size = 180, normalized size = 1.70 \[ \frac {-240 a^3 \tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )-240 a^3 \tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )-2 \sqrt {b} \sqrt {-a-b} \cos (c+d x) \left (120 a^2+4 b (5 a-7 b) \cos (2 (c+d x))-100 a b+3 b^2 \cos (4 (c+d x))+89 b^2\right )}{240 b^{7/2} d \sqrt {-a-b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 272, normalized size = 2.57 \[ \left [-\frac {6 \, {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{5} - 15 \, \sqrt {a b + b^{2}} a^{3} \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) + 10 \, {\left (a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (a^{3} b + b^{4}\right )} \cos \left (d x + c\right )}{30 \, {\left (a b^{4} + b^{5}\right )} d}, -\frac {3 \, {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{5} + 15 \, \sqrt {-a b - b^{2}} a^{3} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) + 5 \, {\left (a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a^{3} b + b^{4}\right )} \cos \left (d x + c\right )}{15 \, {\left (a b^{4} + b^{5}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 332, normalized size = 3.13 \[ -\frac {\frac {15 \, a^{3} \arctan \left (\frac {b \cos \left (d x + c\right ) + a + b}{\sqrt {-a b - b^{2}} \cos \left (d x + c\right ) + \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b^{3}} - \frac {2 \, {\left (15 \, a^{2} - 10 \, a b + 8 \, b^{2} - \frac {60 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {50 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {40 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {90 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {70 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {80 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {60 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {30 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{b^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 110, normalized size = 1.04 \[ \frac {-\frac {\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) b^{2}}{3}+a^{2} \cos \left (d x +c \right )-a b \cos \left (d x +c \right )+\cos \left (d x +c \right ) b^{2}}{b^{3}}+\frac {a^{3} \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{b^{3} \sqrt {\left (a +b \right ) b}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 116, normalized size = 1.09 \[ -\frac {\frac {15 \, a^{3} \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} b^{3}} + \frac {2 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{5} + 5 \, {\left (a b - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a^{2} - a b + b^{2}\right )} \cos \left (d x + c\right )\right )}}{b^{3}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 112, normalized size = 1.06 \[ \frac {a^3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )}{b^{7/2}\,d\,\sqrt {a+b}}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,b\,d}-\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a+b}{3\,b^2}-\frac {1}{b}\right )}{d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {3}{b}+\frac {\left (a+b\right )\,\left (\frac {a+b}{b^2}-\frac {3}{b}\right )}{b}\right )}{d} \]
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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