Integrand size = 25, antiderivative size = 776 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \]
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Time = 0.71 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3294, 1237, 1728, 1209, 1722, 1117, 1720} \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt [4]{b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{2 a d \sqrt [4]{a+b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{2 a d \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt {a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d} \]
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Rule 1117
Rule 1209
Rule 1237
Rule 1720
Rule 1722
Rule 1728
Rule 3294
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\text {Subst}\left (\int \frac {-a+b-2 b x^2+b x^4}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\text {Subst}\left (\int \frac {-b (-a+b)+b^{3/2} \sqrt {a+b}+\left (2 b^2-b \left (b+\sqrt {b} \sqrt {a+b}\right )\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a b d}+\frac {\left (\sqrt {b} \sqrt {a+b}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}+\frac {\left (\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right )\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {\left (\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{a \sqrt {a+b} d} \\ & = -\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.16 (sec) , antiderivative size = 1442, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\sqrt {8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))} \cot (c+d x) \csc (c+d x)}{4 \sqrt {2} a d}+\frac {\sec (c+d x) \left (-a-2 a \tan ^2(c+d x)-a \tan ^4(c+d x)-b \tan ^4(c+d x)-\frac {b \left (\sqrt {b} E\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )+i \left (\sqrt {a}+i \sqrt {b}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (\sqrt {b} \tan ^2(c+d x)+i \sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{(a+b) \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {a \left (-i \sqrt {b} E\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )+\left (\sqrt {a}+i \sqrt {b}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (-i \sqrt {b} \tan ^2(c+d x)+\sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{(a+b) \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {2 a \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (i \sqrt {b} \tan ^2(c+d x)+\sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {i a \sqrt {b} \operatorname {EllipticPi}\left (-\frac {2 i \sqrt {b}}{\sqrt {a}-i \sqrt {b}},\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {(a+b) \left (b \tan ^4(c+d x)+a \left (1+\tan ^2(c+d x)\right )^2\right )}{a b}}}{\sqrt {a}-i \sqrt {b}}\right )}{2 a d \sqrt {\sec ^2(c+d x)} \left (1+\tan ^2(c+d x)\right )^{3/2} \sqrt {\frac {b \tan ^4(c+d x)+a \left (1+\tan ^2(c+d x)\right )^2}{\left (1+\tan ^2(c+d x)\right )^2}}} \]
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\[\int \frac {\csc ^{3}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}d x\]
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\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
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\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
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\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]
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