Optimal. Leaf size=290 \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {\sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{a d (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3900, 3888, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641} \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {\sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{a d (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2573
Rule 2614
Rule 2641
Rule 3476
Rule 3884
Rule 3888
Rule 3900
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx &=\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\int \frac {-a+a \sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {\int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\cos ^{\frac {3}{2}}(c+d x) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{a (e \cot (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\left (\cot (c+d x) \csc (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{a (e \cot (c+d x))^{3/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{a d (e \cot (c+d x))^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{a d (e \cot (c+d x))^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{a d (e \cot (c+d x))^{3/2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {\cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{a d (e \cot (c+d x))^{3/2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] time = 8.86, size = 112, normalized size = 0.39 \[ \frac {4 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \cot ^2(c+d x) \csc (c+d x) \left (\sqrt {\sec ^2(c+d x)}+1\right ) \left (3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\tan ^2(c+d x)\right )+\cot ^2(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{3 a d (e \cot (c+d x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.97, size = 319, normalized size = 1.10 \[ \frac {\left (1+\cos \left (d x +c \right )\right )^{2} \left (i \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-4 \EllipticF \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {2}}{2 a d \left (\frac {e \cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{4}} \]
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 \[ \int \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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 \[ \frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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