Optimal. Leaf size=366 \[ \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-5 b^2\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^3 d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A]
time = 0.60, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2969, 3134,
3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2969
Rule 3081
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (\frac {1}{4} \left (8 a^2-15 b^2\right )-\frac {1}{2} a b \sin (c+d x)+\frac {5}{4} b^2 \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^2 b}\\ &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}+\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{8} b \left (4 a^2-5 b^2\right )+\frac {5}{4} a b^2 \sin (c+d x)-\frac {1}{8} b \left (8 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^3 b}\\ &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\int \frac {\csc (c+d x) \left (\frac {3}{8} b^2 \left (4 a^2-5 b^2\right )-\frac {1}{8} a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^3 b^2}+\frac {\left (-8 a^2+15 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{8 a^3 b}\\ &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (3 \left (4 a^2-5 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{8 a^3}+\frac {\left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{8 a^2 b}+\frac {\left (\left (-8 a^2+15 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{8 a^3 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{8 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{8 a^2 b \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-5 b^2\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^3 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 13.32, size = 435, normalized size = 1.19 \begin {gather*} \frac {\frac {\left (-32 a^2+60 b^2\right ) \cos (c+d x)+4 a \cot (c+d x) (5 b-2 a \csc (c+d x))}{a^3 \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {2 i \left (-8 a^2+15 b^2\right ) \left (-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}}}-\frac {40 a b F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (32 a^2-45 b^2\right ) \Pi \left (2;\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^3}}{16 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1348\) vs.
\(2(435)=870\).
time = 11.46, size = 1349, normalized size = 3.69
method | result | size |
default | \(\text {Expression too large to display}\) | \(1349\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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