Optimal. Leaf size=102 \[ -\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2599, 2601, 2570, 2572, 2639} \[ -\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2570
Rule 2572
Rule 2599
Rule 2601
Rule 2639
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx &=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {2}{5} \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {\left (2 \sqrt {\sin (a+b x)}\right ) \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {\left (4 \sqrt {\sin (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {(4 \sin (a+b x)) \int \sqrt {\sin (2 a+2 b x)} \, dx}{5 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {4 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.68, size = 104, normalized size = 1.02 \[ \frac {6 (\cos (2 (a+b x))-2) \cot (a+b x) \csc (a+b x) \sqrt {\sec ^2(a+b x)}-8 \tan ^2(a+b x) \sec (a+b x) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right )}{15 b \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{3}}{d \tan \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\sqrt {d \tan \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.62, size = 972, normalized size = 9.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\sqrt {d \tan \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^3\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________