Optimal. Leaf size=458 \[ \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-105 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{8 a^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2-35 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {15 b \left (4 a^2-7 b^2\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^5 d \sqrt {a+b \sin (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.98, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2803, 3134,
3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-105 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 )}{8 a^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {15 b \left (4 a^2-7 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 )}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (16 a^2-35 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 )}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2803
Rule 2884
Rule 2886
Rule 3081
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {2 \int \frac {\csc ^3(c+d x) \left (\frac {3}{4} \left (8 a^2-21 b^2\right )-\frac {3}{2} a b \sin (c+d x)-\frac {3}{4} \left (4 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^2 b}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {9}{8} b \left (16 a^2-35 b^2\right )+\frac {27}{4} a b^2 \sin (c+d x)+\frac {9}{8} b \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^3 b}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\int \frac {\csc (c+d x) \left (\frac {135}{16} b^2 \left (4 a^2-7 b^2\right )+\frac {9}{8} a b \left (8 a^2-21 b^2\right ) \sin (c+d x)-\frac {9}{16} b^2 \left (16 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^4 b}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {2 \int \frac {\csc (c+d x) \left (\frac {135}{32} b^2 \left (4 a^4-11 a^2 b^2+7 b^4\right )+\frac {9}{16} a b \left (8 a^4-43 a^2 b^2+35 b^4\right ) \sin (c+d x)+\frac {9}{32} b^2 \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9 a^5 b \left (a^2-b^2\right )}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}-\frac {2 \int \frac {\csc (c+d x) \left (-\frac {135}{32} b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right )+\frac {9}{32} a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9 a^5 b^2 \left (a^2-b^2\right )}+\frac {\left (32 a^4-137 a^2 b^2+105 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{16 a^5 \left (a^2-b^2\right )}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (15 b \left (4 a^2-7 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{16 a^5}-\frac {\left (16 a^4-51 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{16 a^4 \left (a^2-b^2\right )}+\frac {\left (\left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{16 a^5 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-105 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)}}{8 a^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (15 b \left (4 a^2-7 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}{16 a^5 \sqrt {a+b \sin (c+d x)}}-\frac {\left (\left (16 a^4-51 a^2 b^2+35 b^4\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}{16 a^4 \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-105 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)}}{8 a^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2-35 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}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {15 b \left (4 a^2-7 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}}}{8 a^5 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 16.56, size = 680, normalized size = 1.48 \begin {gather*} \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\left (32 a^2 \cos (c+d x)-123 b^2 \cos (c+d x)\right ) \csc (c+d x)}{24 a^5}+\frac {17 b \cot (c+d x) \csc (c+d x)}{12 a^4}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a^3}+\frac {2 \left (a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right )}{3 a^4 (a+b \sin (c+d x))^2}+\frac {8 \left (a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right )}{3 a^5 (a+b \sin (c+d x))}\right )}{d}+\frac {-\frac {2 \left (32 a^3-140 a 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}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (152 a^2 b-315 b^3\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 (-32 a^2 b+105 b^3\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}}}}{32 a^5 d} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2869\) vs.
\(2(519)=1038\).
time = 15.02, size = 2870, normalized size = 6.27
method | result | size |
default | \(\text {Expression too large to display}\) | \(2870\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________