Optimal. Leaf size=408 \[ \frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}+\frac {b \left (236 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{64 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (20 a^2+b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 \left (16 a^4-24 a^2 b^2+b^4\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a^2 d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A]
time = 0.80, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2972, 3126,
3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}-\frac {b \left (20 a^2+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 )}{64 a d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (236 a^2+3 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 )}{64 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}+\frac {3 \left (16 a^4-24 a^2 b^2+b^4\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 )}{64 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2972
Rule 3081
Rule 3126
Rule 3138
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {3}{4} \left (20 a^2-b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {3}{4} \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}-\frac {\int \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {3}{8} b \left (68 a^2-3 b^2\right )-\frac {3}{4} a \left (12 a^2-b^2\right ) \sin (c+d x)-\frac {9}{8} b \left (28 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {9}{16} \left (16 a^4-24 a^2 b^2+b^4\right )-\frac {3}{8} a b \left (108 a^2+b^2\right ) \sin (c+d x)-\frac {3}{16} b^2 \left (236 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{24 a^2}\\ &=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {9}{16} b \left (16 a^4-24 a^2 b^2+b^4\right )-\frac {3}{16} a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{24 a^2 b}+\frac {1}{128} \left (b \left (236+\frac {3 b^2}{a^2}\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx\\ &=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}-\frac {\left (b \left (20 a^2+b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{128 a}+\frac {\left (3 \left (16 a^4-24 a^2 b^2+b^4\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{128 a^2}+\frac {\left (b \left (236+\frac {3 b^2}{a^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}{128 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}+\frac {b \left (236+\frac {3 b^2}{a^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)}}{64 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (b \left (20 a^2+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}{128 a \sqrt {a+b \sin (c+d x)}}+\frac {\left (3 \left (16 a^4-24 a^2 b^2+b^4\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}{128 a^2 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}+\frac {b \left (236+\frac {3 b^2}{a^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)}}{64 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (20 a^2+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}}}{64 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 \left (16 a^4-24 a^2 b^2+b^4\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}}}{64 a^2 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 16.47, size = 641, normalized size = 1.57 \begin {gather*} \frac {\left (\frac {3 \left (36 a^2 b \cos (c+d x)+b^3 \cos (c+d x)\right ) \csc (c+d x)}{64 a^2}+\frac {\left (20 a^2 \cos (c+d x)-b^2 \cos (c+d x)\right ) \csc ^2(c+d x)}{32 a}-\frac {3}{8} b \cot (c+d x) \csc ^2(c+d x)-\frac {1}{4} a \cot (c+d x) \csc ^3(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (432 a^3 b+4 a b^3\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}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (96 a^4+92 a^2 b^2+9 b^4\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}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-236 a^2 b^2-3 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \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 ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{256 a^2 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. \(1759\) vs.
\(2(473)=946\).
time = 13.04, size = 1760, normalized size = 4.31
method | result | size |
default | \(\text {Expression too large to display}\) | \(1760\) |
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(-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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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(-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 {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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