Optimal. Leaf size=73 \[ -\frac {5 \cos (c+d x)}{d \sqrt [5]{\sin (c+d x)}}+\frac {5 \cos (c+d x) \, _2F_1\left (\frac {2}{5},\frac {1}{2};\frac {7}{5};\sin ^2(c+d x)\right ) \sin ^{\frac {4}{5}}(c+d x)}{d \sqrt {\cos ^2(c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 73, 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 = {2868, 2722,
3090} \begin {gather*} \frac {5 \sin ^{\frac {4}{5}}(c+d x) \cos (c+d x) \, _2F_1\left (\frac {2}{5},\frac {1}{2};\frac {7}{5};\sin ^2(c+d x)\right )}{d \sqrt {\cos ^2(c+d x)}}-\frac {5 \cos (c+d x)}{d \sqrt [5]{\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 2868
Rule 3090
Rubi steps
\begin {align*} \int \frac {(1+2 \sin (c+d x))^2}{\sin ^{\frac {6}{5}}(c+d x)} \, dx &=4 \int \frac {1}{\sqrt [5]{\sin (c+d x)}} \, dx+\int \frac {1+4 \sin ^2(c+d x)}{\sin ^{\frac {6}{5}}(c+d x)} \, dx\\ &=-\frac {5 \cos (c+d x)}{d \sqrt [5]{\sin (c+d x)}}+\frac {5 \cos (c+d x) \, _2F_1\left (\frac {2}{5},\frac {1}{2};\frac {7}{5};\sin ^2(c+d x)\right ) \sin ^{\frac {4}{5}}(c+d x)}{d \sqrt {\cos ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 73, normalized size = 1.00 \begin {gather*} -\frac {5 \cos (c+d x)}{d \sqrt [5]{\sin (c+d x)}}-\frac {4 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{5};\frac {3}{2};\cos ^2(c+d x)\right ) \sin ^{\frac {4}{5}}(c+d x)}{d \sin ^2(c+d x)^{2/5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\left (1+2 \sin \left (d x +c \right )\right )^{2}}{\sin \left (d x +c \right )^{\frac {6}{5}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.47, size = 127, normalized size = 1.74 \begin {gather*} -\frac {4\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^{4/5}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{5};\ \frac {3}{2};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\left ({\sin \left (c+d\,x\right )}^2\right )}^{2/5}}-\frac {\cos \left (c+d\,x\right )\,{\left ({\sin \left (c+d\,x\right )}^2\right )}^{1/10}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{10};\ \frac {3}{2};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\sin \left (c+d\,x\right )}^{1/5}}-\frac {4\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^{9/5}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{10},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\left ({\sin \left (c+d\,x\right )}^2\right )}^{9/10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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