Integrand size = 21, antiderivative size = 43 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {2}{5 b d (d \cos (a+b x))^{5/2}}-\frac {2}{b d^3 \sqrt {d \cos (a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2645, 14} \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {2}{5 b d (d \cos (a+b x))^{5/2}}-\frac {2}{b d^3 \sqrt {d \cos (a+b x)}} \]
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Rule 14
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1-\frac {x^2}{d^2}}{x^{7/2}} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{x^{7/2}}-\frac {1}{d^2 x^{3/2}}\right ) \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = \frac {2}{5 b d (d \cos (a+b x))^{5/2}}-\frac {2}{b d^3 \sqrt {d \cos (a+b x)}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.63 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {2 \left (5-4 \sqrt [4]{\cos ^2(a+b x)}+4 \left (-1+\sqrt [4]{\cos ^2(a+b x)}\right ) \csc ^2(a+b x)\right ) \tan ^2(a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {2 d^{2}}{5 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}-\frac {2}{\sqrt {d \cos \left (b x +a \right )}}}{b \,d^{3}}\) | \(37\) |
default | \(\frac {\frac {2 d^{2}}{5 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}-\frac {2}{\sqrt {d \cos \left (b x +a \right )}}}{b \,d^{3}}\) | \(37\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left (5 \, \cos \left (b x + a\right )^{2} - 1\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \]
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Time = 30.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\begin {cases} \frac {2 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{5 b \left (d \cos {\left (a + b x \right )}\right )^{\frac {7}{2}}} - \frac {8 \cos ^{3}{\left (a + b x \right )}}{5 b \left (d \cos {\left (a + b x \right )}\right )^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x \sin ^{3}{\left (a \right )}}{\left (d \cos {\left (a \right )}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {2 \, {\left (5 \, d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )}}{5 \, \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} b d^{3}} \]
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\[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
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Time = 3.46 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.16 \[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {4\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}\,\left (6\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+5\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+5\right )}{5\,b\,d^4\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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